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 to down that year? Perhaps in the last season of TOS someone said onscreen that it was 300 years since Apollo 11?