Continuity mistake: When Uhura tells Kirk that she's made contact with his private transmitter she's seated at the communications console, but after the contact's been broken when she tells Kirk, "It's impossible at the moment because they stopped broadcasting," in this shot facing Uhura the turbo lift is not behind her - she's at a different station on the bridge - then she's back to where she was once again.
Visible crew/equipment: After the Klingon vessel's attack on the Enterprise, while returning fire just as Kirk orders to maintain firing range at 100% dispersal pattern, the set's unfinished wood plank with grey paint drip marks can be seen at the bottom of the screen. Additionally, at the end when Spock goes to stand beside Kirk's chair (also when he walks away) the actor's mark can be seen on the floor by Spock's feet.
Continuity mistake: In the first scene when Spock is piloting the shuttle craft out of the Enterprise and into space, as the bay doors open you can see both the ship's shuttles, the Galileo, and right behind it the Columbus. Yet as soon as the Galileo clears the bay doors, in the immediate reverse shot, the bay is empty and the Columbus has disappeared.
Factual error: When the Enterprise is slingshotting its way back to the "present" it uses the computer to deposit the two men at different times, The Enterprise is traveling backwards in time, so will logically meet the time when the guard was beamed up first, not after Christopher is returned to his cockpit. So the order is wrong. Should be guard then Christopher, not Christopher then guard.