This is a fair episode, and watchable enough the first time or so, but doesn't really have enough appeal in my view to be outstanding. In many ways it's a sort of 'novelty' story, with crew members acting out of character; and doesn't have the depth or the pull of some of the other first season greats.
One of the best and most memorable moments is, of course, Mr. Sulu stalking the corridors, bare-chested and fencing sword in hand (which was immortalised in many episode's closing credits). But special mention has to go to Bruce Hyde as Lt. Riley, who reverts back to his Irish roots and declares himself as new Captain of the Enterprise, sealing himself in Engineering as the ship plunges towards the planet.
Some of the 'out of character' moments (such as the above mentioned Sulu and Riley) work well, but others don't come off as successfully. I know others will fiercely disagree with this, but I find Spock's moments, especially the scene with Nurse Chapel, to be very drawn out and tedious. Spock is one of my favourite characters, but I'm afraid I find that these scenes rather drag.
This story was originally intended to lead into a second part, with the Enterprise having travelled back in time, but this was dropped and it became a stand-alone story (what would have been the second part eventually became 'Tomorrow Is Yesterday'). I think it is a bit of a shame, as the series could have done with a few more two-parters (ultimately, only 'The Menagerie' was the only two-parter of the series).
The story was recycled to serve as a sequel, in 'Star Trek: The Next Generation's early episode 'The Naked Now' (the first episode after the pilot, 'Encounter At Farpoint', of that series). Like 'The Naked Time', that episode doesn't really stand as one of my particular favourites.
All-in-all, this is a watchable episode, but just doesn't really stand out in my view.