We can get very close with clues spoken onscreen throughout the franchise....If we go by that logic we don't have canon years for TNG and probably other shows outside of ENT..
In an episode of Voyager, Icheb said that Kirk's 5-year mission ended in 2270.
Icheb:
"... in the year 2270, Kirk completed his historic five year mission and one of the greatest chapters in Starfleet history came to a close. A new chapter began when Kirk regained command of the Enterprise."
So that gives us a date for TOS. In TNG "Encounter at Farpoint", Data reminds Dr. McCoy that he is 137 years old. From that, and from approximating McCoy's age in TOS (say maybe 45 to 50 years old, give or take, during the five-year mission), then we can get a pretty good idea that TNG began 87 to 90 years after Kirk's first five year mission -- so probably about 2360-ish.
I'm not sure if anything Dax had said onscreen in her reminiscences about McCoy could tighten that estimate up a bit.