Wikipedia notes that ‘Star Trek: The Next Generation’ was set in 2364 (and beyond), and that that’s “about 100 years after the original series timeframe”.
However, the Wikipedia article on The Original Series doesn’t state what year it was set in.
Did The Original Series mention what year it was set in, or were its events later deemed to have occurred in a specific year, e.g. after The Next Generation stated it was set in 2364?
While there's no mention within the Original Series of when it took place, there are a number of clues from other series that narrow down the mission time period.
For example, we know—based on Star Trek: The Next Generation's stardate schedule—that Star Trek: Generations took place in 2371. In it,
Captain Picard remarks to Captain Kirk that he's been dead for 80 years.
Therefore, we know the original mission ended before 2291, and substantially before that given how much occurs after the original mission ends.
We also know that it had to have happened substantially after 2063, which is when First Contact occurred.
TrekGuide.com has a much more accurate conjecture, based on dates and ages from episodes in all the series, but the key fact to me is (emphasis mine):
In the Voyager episode "Q2," Icheb states that Kirk completed his historic five-year mission in 2270. Therefore, all episodes of The Original Series must have taken place after January 1, 2265 (assuming it was exactly five years), and before January 1, 2271.
The original series never stated exactly what year(s) it occurred in. There were a few inconsistent hints.
In "The Squire of Gothos", an early episode, Trelane has been observing Earth as if through a telescope from 900 light-years away. Failing to account for the speed-of-light delay, Trelane reproduces an Earth-like environment approximating the early 19th century or so (he mentions the death of Alexander Hamilton, who died in 1804), implying that the episode takes place in the 27th or 28th century.
In "Tomorrow is Yesterday", a 20th century U.S. Air Force officer threatens to lock Kirk up for two hundred years. Kirk says "That ought to be just about right", implying that the episode takes place in the 22nd century. Kirk's comment was a joke, and it could have been imprecise.
Clearly these are inconsistent, indicating that the producers didn't (yet) have a specific idea about just when the series took place, something that's supported by other sources. Stardates were also not entirely consistent, and there are several different systems for converting between stardates and Gregorian calendar dates.
According to Memory Alpha, the original series took place from 2265 to 2269, almost exactly three centuries after the episodes aired.
In Star Trek (2009), which is set prior to TOS, George Kirk dies/James Kirk is born in 2233. This is consistent with the Memory Alpha TOS dates of 2265-2269 when the junior Kirk would have been about 35.
The 2233 date is pre-timeline-split, so I think it can be regarded as canon for both timelines.
The official Star Trek chronology in Star Trek Chronology: The History of the Future by Michael and Denise Okuda puts Star Trek: The Original series in 2266 to 2269.
This chronology has been followed by the creators of most Star Trek productions from about the early 1990s, so there is a lot of chronological information in various productions which more or less agrees with the dates in Star Trek Chronology: The History of the Future.
However, Star Trek Chronology: The History of the Future is not canonical and thus there is the possibility that it is not correct, as is acknowledged in the introduction, where it is presented as one possible chronology out of many.
In fact it is my belief that it is possible to scientifically disprove the official dates of a couple of Star Trek productions and to disprove many of the assumptions made in Star Trek Chronology: The History of the Future. I hope to someday write a much better chronology.
In ‘The Savage Curtain’ (series 3, episode 22 — the one with Abraham Lincoln and Surak, the greatest ever Vulcan), Scotty says to McCoy that it cannot be the real Lincoln as the real Lincoln died 3 centuries ago.
Given that Lincoln died in 1865 this would mean that Star Trek was set in the mid-22nd century.
I'm partial to the year 2260 as the mission start date.
As fandom emerged, this was the year it circled around. A handful of on screen clues set the date nicely as well, such as the piano in "Miri". (see link for more detail).
I think that this has possibly been contradicted by later canon, but earlier canon and later canon don't always exactly jibe well in the first place.
http://stng.36el.com/st-tng/trivia/timeline/trek7-notes.html#1
I had an opportunity to meet Franz Joseph (the author/illustrator of the Star Fleet Technical Manual and the Star Trek Blueprints) at Starcon 77 (1977), about a month or two after the release of Star Wars: A New Hope.
Franz drew a distinction between a STARDATE and Terrestrial STARDATE when I was visiting with him and his works also have an authenticated STARDATE. He signed my Star Fleet Technical Manual and my Star Trek Technical Blueprints, including a STARDATE for that day of our meeting.
So, in the Star Fleet Technical Manual, Foreword II, there is a copy of a memorandum dated STARDATE 3150.10 from Star Fleet Command to All Divisions Concerned referring to an incident that occurred involving Capt. James T. Kik, commanding U.S.S. Enterprise STARDATE 3113.
The actual Star Fleet Technical Manual has an "authenticated STARDATE of 7309.10" which I believe was meant to me it was actually written in September 10, 1973). This is the format of what he called the tsd (terrestrial STARDATE).
Therefore, as I understand Franz Joseph, Star Trek: The Original Series, was definitely happening during STARDATE 3113.
In "The savage Curtain" Scott said: "Lincoln died three centuries ago on a planet hundreds of light years away, and then Spock corrected him: "More that direction, Engineer." Yes, Spock was picky enough to correct Scott about the direction to Earth but not the time since Lincoln died.
We might assume that 3 centuries could be as vague as 200 to 400 years, putting "The Savage Curtain" sometime between 2065 and 2265. The list of assumptions in the introduction to Star Trek Chronology: The History of the Future says the Okudas who wrote it assumed that X centuries would always be exactly X hundred years, putting the episode in 2165.
This assumption of course invalidates the official date of 2269 for "The Savage Curtain". Even assuming that "The Savage Curtain" could be as much as 400 years after 1865 without Spock correcting Scott could only make it as late as 2265, still years before the official date.
This shows how carelessly Gene Roddenberry, Richard Arnold, and the Okudas chose whatever dates they wanted, without bothering to check if those dates were consistent with the evidence in the episodes.
To bad cause I just found some link to proove my answer but it is only in french, the english version of this page is not really accurate compared to this french version filled with links.
2265 :
Start of the 5 years mission of the Enterprise NCC-1701 under the commandment of Captain James T. Kirk.
Hikaru Sulu is affected as a physicist on Enterprise NCC-1701.
Montgomery Scott is affected as chief engineer on Enterprise NCC-1701.
Doctod Mark Piper becomes the chief medical officer on Enterprise NCC-1701 succeding Philip Boyce.
So the begining of the 5 years mission with Captain James T. Kirk at the command of The Enterprise NCC-1701 is 2265
For any more information regarding dates from the star trek univers !
