Factual error: Kirk's line, "We're a doomed ship, traveling forever between galaxies" is the only time original "Trek" committed the scientific blunder of confusing galaxies and solar systems. The Enterprise wasn't capable of intergalactic travel (that's leaving one galaxy and reaching another). Yes, it strayed briefly out of our galaxy several times. But it did not - and could not - cross to another one. That would take a warp 100-plus drive and thousands of years.
Revealing mistake: When Kirk and Spock answer the intercom in the ship's corridor, Spock's uniform has a large brown coffee stain on the front. (This was digitally removed in the DVD release, but is still visible in the newly-syndicated "computer enhanced" version.).
Other mistake: At the end, Kirk teases Spock twice specifically about letting himself be hit on the head in order to determine which Kirk was genuine. But the phony Kirk never hit Spock on the head. He merely pushed Spock over and attacked the real Kirk. Clearly, the action sequence wound up differing from the scripted dialogue, and no one noticed. (Also clearly, Kirk is not speaking metaphorically here.)
Continuity mistake: While Kirk and McCoy attack the other two henchmen, Spock shoulder-pinches the card-playing gangster. In the very next shot, Spock's position has jumped from behind the now-unconscious man to several feet away, where he is bending over to retrieve the guns from the floor.
Continuity mistake: Mr. Leslie appears as one of the guards in the space station, Stardate 45XX.X. Unfortunately, he was killed in 'Obsession,' Stardate 36XX.X. External websites have a variety of excuses for this miraculous reappearance, but he very definitely dies in the show, with no canonical reason for him reappearing beyond it being a mistake.
Continuity mistake: In a few episode we'll hear about "Hodgkin's Law of Parallel Development," but here Spock claims the odds of a planet developing similar to Earth's Nazi culture are "virtually impossible." Why was it so likely as to be easily accepted there, but impossible here?
Continuity mistake: In the very first scene after the intro, when Bela's goons hold the landing party at gunpoint, Spock's position on the sidewalk shifts back and forth from standing on the other side of a wooden bench to standing alongside Kirk and McCoy with no bench in between. In fact, in the latter shots, the bench has moved all the way across the street.