I remember the idea that the Galaxy-class was that design.
The idea could also be that it's Galaxy class, not Galaxy class. That is, since all the ships are named after galaxies, it's analogous to the various Royal Navy "Town classes" or "County classes" that never feature an HMS Town or an HMS County but merely utilize thematic names.
Not that I'd be happy with these ships actually visiting faraway galaxies within a few years. Nearby dwarf galaxies, yes, and perhaps within a decade or two. So...
Given that ships move at the speed of plot, just how long can a mission be in terms of time?
Three decades might well be doable. There would be some deaths or retirements among the crew, but perhaps also some births leading to new crew serving for a decade or more towards the end of the mission. But the ship in question might need to have a bit more room and amenities than Kirk's TOS ride, and perhaps a bit more self-repair capacity. Still doable with TOS technology, I'd wager.
Timo Saloniemi