I forgot.... Doesn’t even nuclear powered craft need to be refueled every 20 or so years?
Current designs, yes. A century design is certainly possible.
Nuclear warheads need periodic maintenance as well.
Targeting will need updates (Who were we at war with 100 years ago? Spain was it?).
Maybe slightly longer than that, but it depends on how hard you run 'em. Besides they aren't really refueled anymore since that would involve cutting into the pressure hull. They just de-fuel and scrap them when the reactor cores wind down.