Did you ever hear of Isaac Asimov's "galactic empire" trilogy as it is sometimes called?
Anyone who has read the three novels will realize that The Stars Like Dust must be many thousands of years before The Currents of Space which should be thousands of years before Pebble in the Sky.
But as I remember, in his introduction to Prelude to Foundation Asimov listed the fictional order of those three as The Currents of Space, The Stars Like Dust, and Pebble in the Sky.
And when I read that I simply figured that Asimov (or a typesetter) had made a simple error in listing the novels. I believed that when and if Asimov realized that he would correct the order in future lists of the fictional order of his stories.
I never thought for a second that Asimov might have thought about it and decided to change the fictional order of The Stars Like Dust and The Currents of Space, because there was no logical, rational way for the events in The Currents of Space to happen before the events of The Stars Like Dust.
The writer of a story is like a creator god of the fictional universe of the story, with immense power over it. But the writer's power over his story is not infinite, any more than the power of the Greek gods or the Norse Gods was infinite in the myths about them. The writer of a story can not arbitrarily decree illogical, inconsistent, irrational, and mathematically impossible details for his fictional world.
And I considered the official dates in Star Trek Chronology: The History of the Future, to sometimes be too illogical, inconsistent, irrational, and mathematically impossible to be "correct" in the fictional universe of Star Trek.
In the second pilot film "Where No Man Has Gone Before", there are stardates and also three other sets of dates, two of which might be either stardates or calendar dates.
In the sickbay:
MITCHELL: My love has wings. Slender, feathered things with grace in upswept curve and tapered tip. The Nightingale Woman, written by Phineas Tarbolde on the Canopius planet back in 1996. It's funny you picked that one, Doctor.
MITCHELL: That's one of the most passionate love sonnets of the past couple of centuries. How do you feel, Doctor?
If 1996 was during the past couple of centuries the current date must be between 1996 and 2196.
Kirk's first log says:
Captain's log, Star date 1312.4. The impossible has happened. From directly ahead, we're picking up a recorded distress signal, the call letters of a vessel which has been missing for over two centuries. Did another Earth ship once probe out of the galaxy as we intend to do? What happened to it out there? Is this some warning they've left behind?
If the date is between 1996 and 2196, the SS Valiant lost over two hundred years earlier must have been lost in the period of 1796 to 1996, or earlier.
The medical files of Gary Mitchell and Elizabeth Dehner say Mitchell was born 1087.7 and give his age as 23, and Dehner was born 1089.5 and give her age as 21. If their ages are given in years of a calendar, then the current date should be 1110.7 to 1111.5. Of course their ages might be their ages when they were commissioned in Starfleet years ago. In any case their probable age range means the year should be long before 1996 to 2196, and also long before 1313.
Mitchell makes a tombstone for Kirk when he prepares to kill him. The stone says: "James R Kirk C 1277.1 to 1313.7". The dates might be might be the dates of Kirk's life or the dates that Kirk was in command of the Enterprise.
So this can be interpreted to mean that three different groups of dates are given in "Where No Man Has Gone Before", dates that seem to be given in three different calendars. But it is possible that the dates in Mitchell's and Dehner's records and/or the dates on the tombstone are given in stardates, would would be a problem for interpreting stardates instead of a calendar problem.
And the date of "Where No Man Has Gone Before" seems to be between 1996 and 2196.
On the street of an abandoned town in "Miri" with Earth like architecture:
KIRK: Identical. Earth, as it was in the early 1900s.
SPOCK: More the, er, mid-1900s I would say, Captain, approximately 1960.
In a house they see a piano (the actual physical piano must have made in 1966 or earlier, of course):
KIRK: How old is this thing?
SPOCK: About three hundred years.
And some fans like to imagine this means Kirk and Spock were talking 300 years after 1966. But Kirk didn't ask "When did Earth make pianos with this design?", he asked either "How long ago was this piano manufactured?" or "how long has this piano been abandoned?", and so the answer should be either that the wood has been dead for about 300 years or else that it has been a gathering dust for about 300 years.
So this says nothing about how much time has passed since AD 1966.
In "Shore Leave" Sulu finds a gun:
(They find Sulu target practising)
KIRK: What do you think you're doing?
SULU: Target shooting, Captain. Isn't it a beauty? Haven't got anything like this in my collection.
KIRK: Where did you get it, Mister Sulu?
SULU: I found it. I know it's a crazy coincidence, but I've always wanted one like this. Found it lying right over there. An old-time police special, and in beautiful condition. Hasn't been one like this made in a couple of centuries.
So "Shore Leave" should be 150 to 250, or maybe even 100 to 300 years, after pistols like that stopped being made.
In "Tomorrow is Yesterday" the Enterprise is hurled to Earth in the past.
KIRK: Manned Moon shot? That was in the late 1960s.
SPOCK: Apparently, Captain, so are we.
The late 1960s would be the years 1965 to 1969 in the calendar used.
When Kirk is interrogated by Colonel Fellini:
FELLINI: I am going to lock you up for two hundred years.
KIRK: That ought to be just about right.
This implies that TOS should be about 100 to 300 years in the future, allowing for normal imprecision, and thus in the period 2065 to 2269.
In "The Gallileo Seven" Latimer is speared by a giant native:
SPOCK: (pulling the bloody spear from Latimer's body and examining it) Folsom Point.
SPOCK: There's a remarkable resemblance to the Folsom Point discovered in 1925, old world calendar, New Mexico, North America. A bit more crude about the shaft, I believe. Not very efficient.
But online sources say that the Folsom point was discovered in AD 1926. Therefore Star Trek must be in an alternate universe where the Folsom Point was discovered in 1925 instead of 1926, or else in an alternate universe where the "old world Calendar" has a year one in AD 2 instead of AD 1.
In "The City on the Edge of Forever", Spock, Kirk, and McCoy find themselves in New York City in the year 1930.
MCCOY: The only possible answer would conclusively prove that I'm either unconscious or demented. This looks like old Earth around 1920 or 25.
EDITH: Would you care to try for 30?
So the year is 1930 in Edith's calendar.
Later Edith says:
EDITH: We can talk about that later. I have to go. My young man is taking me to a Clark Gable movie.
EDITH: If we hurry, maybe we can catch the Clark Gable movie at the Orpheum.
According to IMDB, Clark Gable hadn't been a credited actor in a movie since North Star in 1925, and all of his roles were as un credited extras until 1931 when he had credited roles again and even leading roles.
So either Star Trek is in an alternate universe where Clark Gable became a star by 1930, or else Star Trek is in an alternate universe where the calendar in Edith Keeler's era had it's year one in AD 2 or later.
In "The Squire of Gothos" Trelane's knowledge of Earth seems centuries outdated and Jaeger and Kirk speculate about the possible cause:
TRELANE: I can't tell you how delighted I am to have visitors from the very planet that I've made my hobby. Yes, but according to my observations, I didn't think you capable of such voyages.
JAEGER: Notice the period, Captain. Nine hundred light years from Earth. It's what might be seen through a viewing scope if it were powerful enough.
TRELANE: Ah, yes. I've been looking in on the doings on your lively little Earth.
KIRK: Then you've been looking in on the doings nine hundred years past.
TRELANE: Oh, really? Have I made an error in time? How fallible of me. Oh, I did so want to make you feel at home. I'm quite proud of the detail.
TRELANE: Ah, monsieur. Vive la gloire. Vive Napoleon. You know, I admire your Napoleon very much.
General Napoleon Bonaparte became First Consul in 1799, Consul for life in 1802, and was proclaimed Emperor Napoleon I on 18 May 1804.
Later Trelane gets a pair of dueling pistols:
TRELANE: A matched set. Just like the pair that slew your heroic Alexander Hamilton. And Captain, I never miss.
Alexander Hamilton was fatally wounded in a duel on 11 July, 1804.
So if "Squire of Gothos" must be at least "900 years" (800-1,000 allowing for human vagueness) after 1804 the date must be at least in the period 2604 to 2804.
That date range is centuries after the official date of "Squire of Gothos" and more importantly is centuries after the date ranges suggested by most of the other evidence.
And since looking at Earth through a super telescope couldn't help Trelane learn Earth words or names or Earth music, Trelane should have used some other method to learn about Earth and there should be another reasons why his knowledge was centuries out of date.
In "Space Seed", at first they think that Khan and the Botany Bay are from the early or mid 1990s:
SPOCK: No such vessel listed. Records of that period are fragmentary, however. The mid=1990s was the era of your last so-called World War.
KIRK: What was the exact date of your lift off? We know it was sometime in the early 1990s, but
On the Botany Bay:
KHAN: How long?
KIRK: How long have you been sleeping? Two centuries we estimate. Landing party to Enterprise. Come in.
And in the Enterprise sickbay:
KHAN: I remember a voice. Did I hear it say I had been sleeping for two centuries?
MCCOY: That is correct.
KIRK: Yes, I understand. You have two hundred years of catching up to do.
Assuming that the early 1990s is from 1990.00 to 1993.33, and that 200 years is 100 to 300 years allowing for human imprecision, the date of "Space Seed" should be about 2090.0 to 2293.33. Thus the voyage of the Valiant from "Where No Man Has Gone Before" would be in the period between about 1890.0 and 2093.33.
So to sum it up:
"Where No Man Has Gone Before" should be between 1996 and 2196.
"Tomorrow is Yesterday" should be between 2065 and 2269.
"The squire of Gothos" might be between 2604 and 2804, except that would conflict with the other date ranges.
"Space Seed" should be between 2090.0 and 2293.33.
And the combined date ranges from "Where No Man Has Gone Before", "Tomorrow is Yesterday", and "Space Seed" give a date range for TOS between 2090.0 and 2196.
So those are the date indications I was able to find in the first season episodes.