The Original Series is set from 2265 to 2269, but as far as I can remember, the year was never given until much later, in The Next Generation, which arbitrarily started 100 years later.
Somehow, someone, most likely Mike Okuda or Gene Roddenberry himself, decided to make 2265 the start year, but how was that decided? What specific clue or line tied it down to that year? Perhaps in the last season of TOS someone said onscreen that it was 300 years since Apollo 11?
It seems my comment above was incorrect on two accounts. First, the direct answer to the question. On the Memory Alpha page for Stardates is an excerpt from an interview with Gene Roddenberry, including these lines:
In the beginning, I invented the term "star date" simply to keep from tying ourselves down to 2265 A.D., or should it be 2312 A.D.? I wanted us well into the future but without arguing approximately which century this or that would have been invented or superseded.
The book it appeared in was thought up in May 1967 and published in September 1968 - so Roddenberry had the year 2265 in his mind as early as Season 2 of The Original Series.
Here's the other part, that surprised me, from higher up on the same page: Apparently the fact checker had his own opinions on the very first episode. He thought "Stardate 1312.4" should instead be renamed to "Julian B 1312.4", a time system based on the real-life Julian days. That particular date would translate to August 5, 3271.
The script writers may well have used that Julian day system to come up with the inconsistent "200 or 1200 years in the future" that TOS kept falling to. Which sort of worked to Roddenberry's advantage, because as he said in the interview, he didn't want to quibble over the exact year.
(And I have to say, they kept the year confusing pretty well!)
The introduction to the 1993 edition of Star Trek Chronology: The History of the Future, Michael Okuda and Denise Okuda page v, says:
Basic assumptions: This chronology is built on a number of basic assumptions. The first is that the original Star Trek series was set 300 years in the future of the first airings of the episodes, meaning that the first seasons was set in 2266-67. Although a few references exist suggesting the producers of that show vacillated between 200 to 800 years, the 300 year figure seems to be the most internally consistent...The second basic assumption is that the first season of Star Trek: The Next Generation was set in 2364, as established in "The Neutral Zone". These dates have been used by the producers of Star Trek: The Next Generation and some of the writers of the Star Trek features, so many ages and dates mentioned in the show have been consistent with these assumptions. Most of the dates set forth in this chronology were derived from these two basic assumptions.
And on page iv, discussing Michael Okuda's first efforts at forming a chronology for the use of Star Trek writers:
Former Star Trek research consultant Richard Arnold proved to be a tremendous help at this stage, providing for us (as he had for some of Star Trek: The Next Generation's writers) many of the basic assumptions that form the framework on which this chronology is built.
Thus I deduce that Richard Arnold and Gene Roddenberry decided on many of the basic assumptions which were used by the Okudas in their official chronology, and very probably arbitrarily decided that TOS episodes happened 300 years after being aired, and that TNG's first season would be 400 years after production began on The Cage.
By arbitrarily deciding on those dates Roddenberry and Arnold chose not to support the earliest publish Star Trek chronology from Star Trek: An Analysis of a Phenomenon in Science Fiction (1968), nor the chronology in Star Trek technical fandom, nor the chronology in Star Trek Spaceflight Chronology, nor any other previous chronology.
By arbitrarily deciding on those dates Roddenberry and Arnold and the Okudas omitted checking to see if their dates were possible, and so chose dates which can be shown to be impossible according to evidence from previous Star Trek productions.
For example, as early as TOS the use of dates in several different Earth calendars was seen, proving that several different Earth calendars were used in Kirk's era, and the first couple of seasons of TNG continued the use of different Earth calendars.
And yet the Okudas, like almost everyone else, continued to make the unspoken assumption that all Earth dates are given in one calendar, our present Gregorian calendar. And, like almost everyone else, they failed to notice that if all the Earth dates were given in the Gregorian calendar, Earth would have had to discover FTL travel twice.
The year seems to be have been decided in Star Trek the Motion Picture (1979), where it is discovered that "V'ger" was in fact the Voyager probe, which Decker pointed out was "launched more than 300 years ago". I'm not sure if the precise year is mentioned in the film, but Wikipedia gives the year as 2171.
