There are two basic ways to answer this question--first in terms of the conscious intent of the writers and designers on TOS, second in terms of what later canon established about the TOS-era Enterprise. I'll add a little to what @Tango said about the first question, and then also address the second question.
How the creators imagined things at the time
Early on in the production of the show, the creators hadn't even developed the idea that the ship was powered by matter/antimatter reactions. In the behind-the-scenes book The Making of Star Trek, written while the third season of TOS was still in production, there is this comment on p. 192:
The matter-anti-matter engines were not fully settled upon until after the series was already on the air. Originally the Enterprise was said to be powered by something loosely called a "space warp." As episode after episode was into production, it became increasingly obvious that this point would have to be tied down.
Discussions with scientific consultants had already ruled out atomic power as inefficient and inadequate for achieving hyper-light speeds. Ion drive was ruled out for the same reasons.
Finally, the conclusion reached was that the only power source conceivably large enough to do the job would be the energy released by the sheer annihilation of matter and anti-matter.
Even after the idea of a matter/antimatter engine was developed, there doesn't seem to have been a notion that there was an engine in the secondary hull (the part below the saucer section) which supplied power to the nacelles, as was established in later incarnations of Trek. Instead the idea seems to be that the power-generating warp engines were located within the nacelles themselves. Page 62 of the book Star Trek: The Original Series Sketchbook quoted Enterprise designer Matt Jefferies on why he put the nacelles away from the rest of the ship:
"I was concerned about the design of a ship that Gene told me would have 'warp' drive. I thought, 'What the hell is warp drive?' But I gathered that this ship had to have powerful engines—extremely powerful. To me, that meant that they had to be designed away from the body."
Likewise the third revision of the guide for writers on the show from April 1967, written by Gene Roddenberry and expanded by D.C. Fontana and available online as a pdf here, says on p. 8:
The Enterprise engines (the two outboard nacelles) use matter and anti-matter for propulsion, the annihiliation of dual matter creating the fantastic power required to warp space and exceed the speed of light.
The second season episode "By Any Other Name" also has Scotty referring at one point to the "matter-anti-matter nacelles", which also suggests the idea that the matter/antimatter power-generating system was located within the nacelles. And in another second season episode, "Bread And Circuses", they are examining the wreckage of the "survey vessel SS Beagle", and Spock identifies "Portions of the antimatter nacelles". Later on in the Animated Series episode "One Of Our Planets Is Missing", Kirk says "We need both the matter and the antimatter engines regenerated" suggesting they were imagined as separate, and Scotty suggests a plan where they capture an external chunk of antimatter and "take it into the antimatter nacelle", possibly suggesting the idea that one nacelle used matter and the other used antimatter. (Gene Roddenberry and D.C. Fontana were executive producers on the animated series, and this interview suggests that Fontana and TOS writer David Gerrold both took the show seriously as a continuation of the original live-action series.)
TOS also established that dilithium crystals played some kind of key role in the power system. In some of the earliest episodes of season one these were referred to as "lithium crystals". In the early first-season episode in the episode "Mudd's Women", the ship's engines are overloaded and Scotty explains a sudden flickering of the lights by saying "That was one of our lithium crystal circuits" and then after another lithium circuit goes, he says "Now supplementing with battery power, sir." When all but one crystal has ceased functioning, Spock says "The entire ship's power is feeding through one lithium crystal" and Scotty says "it's frustrating. Almost a million gross tons of vessel depending on a hunk of crystal the size of my fist", and Spock responds "And that crystal won't hold up, not pulling all our power through it." So here it sounds like power was not generated by the crystals, but needed to pass through it, perhaps to change from one form to another.
But later in the first season, the episode "The Alternative Factor" seemed to treat the crystals (now called 'dilithium' crystals) as a power source, with a Lieutenant named Masters saying that a strange phenomenon had "drained almost all of our crystals completely" and Kirk responding "Without full crystal power, our orbit will begin to decay in ten hours". Later when Lazarus says that his "enemy" has stolen some of the crystals, Spock asks "Why does he need the crystals?" and Lazarus responds "He can steal an energy source for his vehicle so he can escape from me." Later Lazarus says that his crystals have been "drained" but that "It'll take him about ten minutes to re-energise with the equipment aboard his ship." It's possible of course that the writers did intend this to be consistent with the idea in "Mudd's Women" that power just passed through the crystals in some way that was essential to the functioning of the engine, but that they needed to be "energized" in some way to work properly, and that Lazarus referring to them as an "energy source" can be chalked up to sloppy writing or the writers being a bit hazy on the distinction between the two. The notion that the dilithium crystals need to be energized or charged in some way to function is also seen in the third season episode "The Day Of The Dove", where Scotty at one point says "The ship's dilithium crystals are deteriorating" and later says "There's no change, Captain. The dilithium crystals are discharging."
Warp equipment in TOS era as understood in terms of later canon
In the TNG era, the basic picture of how the warp engine works is explained in the Star Trek: The Next Generation Technical Manual, which says that the matter and antimatter were channeled through the dilithium crystal, where they would react in a non-explosive way somehow made possible by the unique nature of dilithium, and the high-energy plasma that resulted from their reaction would be channeled to the warp nacelles.
If we retroactively assume the warp drive in the TOS era worked the same way, then according to the memory alpha article Warp Core, one part of the warp core was visible in the TOS era engineering section--the "dilithium crystal converter assembly", "which consisted of two flattened rounded nodules situated directly in front of the warp plasma conduits to the warp engines, which were behind a large metal grate". The article on the assembly shows a shot of this part of the warp core, from the episode "Elaan of Troyius" (where Scotty actually used the term 'dilithium crystal converter assembly'):
Note that although in this scene the chamber with the dilithium crystal is exposed, most of the time it was retracted between the "two flattened rounded nodules" described above, which you can see better in this shot from "Day of the Dove":
There's a photo in the U.S.S. Enterprise Owners' Workshop Manual (which was written in consultation with Michael Okuda, who with Rick Sternbach was the technical consultant on all the shows from TNG onwards, so I think it can be considered to have almost the same level of canon as the TNG Technical Manual which was written by Okuda and Sternbach) which shows these two rounded things and calls them the "dilithium-focused matter/antimatter integrators" (the rounded things are not referred to as matter/antimatter integrators in any onscreen dialogue, though in the episode "That Which Survives", an engineer named Watkins points at a control panel on the wall and says "This is the matter-antimatter integrator control"). The text also says "Before 2267, the dilithium crystals were located in a separate control room. However, during a layover at the end of 2267 the Enterprise NCC-1701's main engineering room was redesigned and the twin matter/antimatter integrators were fitted in the centre of the room. These redesigned units contained the dilithium crystal converter assembly." Presumably the out-of-universe explanation for this is that they redesigned the engineering set somewhat, around the time of one of the later episodes that have since been defined to have taken place in 2267--this site on the history of changes to the engineering set suggests it may have been around the time of the second-season episode "The Doomsday Machine".
The article here also proposes an alternate possible explanation for the lack of the matter/antimatter integrators in earlier episodes: the Enterprise could have had two different engineering rooms, with one located "at the rear of the Primary Hull with the Impulse Drives" and the other "nestled in the Secondary Hull, where the Movies place it." A diagram of the putatively distinct engineering rooms (originally from Star Trek Fact Files as seen in the image gallery here, though I don't think the magazine suggested they were two different engineering rooms, just the same set dressed differently in earlier vs. later seasons), is shown below.
As for the rest of the warp core, where the matter and antimatter would be channeled up towards the dilithium crystals, nothing like this was shown onscreen in TOS, but there's a note in the Warp Core article saying:
Of the original Constitution-class warp core, only the dilithium crystal assembly and the plasma conduits were ever seen in Star Trek: The Original Series. When Doug Drexler was called to design the detailed schematics of a Constitution-class starship, he designed a horizontal warp core that runs two decks below main engineering. The schematic made a prominent appearance on screen in ENT: "In a Mirror, Darkly, Part II".
Here is the schematic of a Constitution class ship designed by Doug Drexler (also mentioned in Bill Teel's answer). And below is the same image with labels, from an archived copy of Doug Drexler's old "Drex Files" site (linked on the memory alpha Constitution class decks article, which has a more detailed guide to what's what)--the stackexchange software shrunk the image when I uploaded it, but you can see the full size version here (click it to enlarge).
This schematic did appear onscreen in "In a Mirror, Darkly, Part II" so it's canon to that extent, although there were no labels so there isn't really any clue that the yellow line labeled "Warp Core" above is supposed to be part of the warp engine:
Looking at the labeled version, if you zoom in on the section that has "Main Engineering" and "Warp Core", you can see that the matter-antimatter integrators with the two rounded tops are shown in the engineering section, and they sit atop some kind of vertical shaft in the floor below, which meets the horizontal shaft labeled "Warp Core" in some kind of central structure--it's unclear from the diagram whether that label is only meant to apply to the horizontal shaft, or to combination of the vertical and horizontal shafts and the structure where they meet.
I think it's likely Drexler meant this to recall the design of the warp core seen in the refit Enterprise from Star Trek: The Motion Picture, which similarly had a vertical shaft meeting a horizontal shaft:
According to the officially licensed tie-in book Mr. Scott's Guide to the Enterprise, this is the intermix chamber. (It's not referred to by this name in the movie, though in this scene Scotty does say 'Intermix set, bridge. Impulse power at your discretion.') Page 86 of the book says this is located in the Engineering section on "O Deck", whose location is shown in a cutaway diagram on p. 20:
So according to this diagram it's level with the top of the deflector dish rather than the bottom as in Drexler's diagram, but this could be imagined as part of the refit, or just chalked up to Mr. Scott's Guide not being fully canonical (though as @O.R.Mapper points out in a comment, toward the back of the horizontal shaft we see two conduits extending diagonally upward, which would fit with the idea that these are meant to supply power through the two pylons that support the nacelles, and in external views of the ship you can see those pylons do join the secondary hull at around the same level as the top of the deflector dish).
Page 88 describes the intermix chamber, saying that the vertical shaft provides power to the impulse drive and other shipboard systems, while the horizontal shaft provides power to the warp field nacelles and phasers:
Located in the center of the room, and extending for many levels both above and below the deck, is the vertical linear intermix chamber. This complex, radically new design in intermix technology provides operational power for the impulse drive system and furnishes enough additional energy to power all other shipboard systems. Both matter and antimatter for the chamber are contained in a series of magnetic bottles, which are housed in pods at the base of the intermix shaft. These pods may be ejected from the ship in case of extreme emergency via two large blow-away panels in the outer hull.
Extending aftward from the vertical shaft is a horizontal chamber which draws its matter/antimatter fuel from the same source. This shaft provides source energy for the warp field nacelles and phaser banks. The linear configuration has proven to be consistently cooler, cleaner, and more efficient than any other system in use today.
The horizontal intermix shaft extends for one hundred forty-five feet aftward, at which point it branches out and upward toward the nacelle pylons.
In the TNG technical manual the basic design of the warp core (also known as the matter/antimatter reaction assembly) was that matter and antimatter flowed through the vertical column to the matter/antimatter reaction chamber containing the dilithium crystals at the center, which as mentioned earlier controlled the matter/antimatter reaction and converted them into warp plasma, which was then directed to other ship systems to power them, including the warp nacelles. The description from Mr. Scott's Guide to the Enterprise makes it sound like they hadn't quite settled on this picture of how the warp engine works at the time of the The Motion Picture, but if we want to retcon it to make it fit with TNG, we might imagine that the vertical and horizontal shafts contain separate internal channels for matter, antimatter, and warp plasma, with the matter and antimatter directed inwards to the dilithium crystals, and the plasma directed outward through the same shafts towards different ship systems including the nacelles. In that case, the two shafts in Drexler's diagram of the TOS era engine could have worked the same way.
Page 88 of Mr. Scott's Guide mentions the dilithium was housed in the "dilithium reactor room" which was on the same deck as the spot where the two intermix shafts met, but it also said this was a "recent addition to Enterprise's warp engineering section", so it could be imagined to be an updated version of the matter/antimatter integrators seen in TOS. And while the dilithium reactor room wasn't built right on top of the vertical intermix shaft as the matter/antimatter integrators seem to be in Drexler's diagram, it was directly adjacent to it (so perhaps we can imagine some short conduits under the floor which connected the reactor room to the junction between the vertical and horizontal intermix shafts, with matter and antimatter flowing to the dilithium reactor and plasma flowing back to the junction). The photo of the dilithium reactor room on p. 89 shows it to be the room where Spock received the fatal dose of radiation in The Wrath of Khan, and you can see in the scene starting 57 seconds into the clip below that the vertical intermix shaft is right behind Kirk when he goes to have his last talk with Spock:
Finally, the Memory Alpha "Warp Core" article claims that a vertical warp core was seen in the animated series:
Star Trek: The Animated Series also featured a vertical component of the warp core, that extended from the dilithium crystal assembly.
Here is an image of the vertical column in engineering that they're referring to (it seems to be a composite of different shots in the animated series, but the vertical column can be seen in the episode "One of Our Planets Is Missing"):
I haven't found any onscreen or licensed source that says this was supposed to be a warp core, so this may have just been some wiki editor's fan theory. Two episodes of the animated series, "One of Our Planets is Missing" along with "Beyond the Farthest Star", make reference to something called the "engineering core" which was where the "self-destruct device" could be armed, but having watched both episodes, there is nothing to connect the engineering core to that vertical column. Note however that the base of the column does look nearly identical to the base of the rounded structures that later authors called "matter/energy integrators"--compare it with the second image in this answer (the one from "Day of the Dove"), the bases both look like triangles with the corners cut off with three rounded parts near the cut-off corners. So that would at least make it plausible that whoever drew this structure intended it as an updated version of the structure seen in TOS, which was shown onscreen to house the dilithium crystals. If you want to treat the animated series as having some level of canonicity, it might then make sense to assume this structure was an extension of the vertical shaft seen in Drexler's diagram, perhaps containing the dilithium crystals somewhere within.