One must remember that the USS Enterprise during Kirk's five year mission, was already an old ship. We know this at the very least by witnessing an adventure of the ship under the command of Captain Pike over a decade prior to Kirk being show it. The Klingons consider her old as well and use that to taunt Mr. Scott.
If we go by the older assumed commissioning year of 2245, than the USS Enterprise is pretty much 25 years old when Kirk brings her back in from his five year mission after the end of TOS and probably TAS. She than gets a major refit (more like a reconstruction really) and goes back into service after the V'Ger Incident until her destruction over a decade later. At roughly 40 years old Starfleet seems to wish to retire the old starship, we assume because they want to use her name on a new Excelsior-class starships in the near future.
After her destruction, Kirk is granted another starship of the same class named USS Enterprise, but it is still debated on if she is a new ship or a renamed older ship, as there is evidence to support both claims, and the oddity of her supposed retirement after less than a decade of service.
We are unsure just how old the class really is as we don't know for sure when USS Constitution was built, nor if it is even the class ship, or just the subclass ship that the rest later were generally called by in the following decades. Starship-class being weird, but would fit a British mind set.
