Space Seed - S1-E23
Continuity mistake: At the hearing in the final scene, there are two bits of debris of some sort (small, black and roundish) littering the floor at the bottom left of the screen. Whatever it was disappears when Scott, Spock, Kirk and McCoy get up to leave the room. (00:48:20)
Space Seed - S1-E23
Continuity mistake: When the landing party beams over at the beginning, the transporting officer is wearing a blue jumpsuit, but when they go to the closeup of his hands, they have a red uniform, two braid stripes, and are the standard shot they use when Scotty does the transporting. (00:05:05)
Space Seed - S1-E23
Plot hole: It is stated during the episode that Khan was, at one time, the ruler of over 1/4th the Earth's population, during a very key moment in the planet's history (The "Eugenics" Wars). Such a personage undoubtedly would be very well known to 23rd century Earth people, at a level of infamy approximating Julius Caeser or Adolf Hitler. Yet it is only 2/3rds the way through the episode, thanks to a computer search by Spock, that the crew divines his identity. Lt. McGivers at the very least should have almost instantly recognized him.
Space Seed - S1-E23
Other mistake: Spock calls for McGivers and Scotty to the transporter room. In the Transporter Room, Scotty is at the controls, then after he went to the Transporter pad, they used a stock shot of his hands working the controls.
Space Seed - S1-E23
Revealing mistake: During the fight sequence between Kirk and Khan in the engine room, you can see in some shots that it is not William Shatner and Ricardo Montalban fighting, but their stunt doubles.
Space Seed - S1-E23
Visible crew/equipment: When Khan hides behind the wall in the engine room before Kirk enters, a crew member walks in front of the stage light and you can see the shadow of his head passing across the floor.
Space Seed - S1-E23
Visible crew/equipment: When the landing party are aboard the Botany Bay talking about it being a sleeper ship, Kirk asks if it's possible they're still alive, and as the camera follows Kirk the shadow of the camera/cameraman is visible on the wall, at the right side of the screen. (00:06:25)
Space Seed - S1-E23
Continuity mistake: As the away team observes Khan in the stasis chamber, McCoy holds his tricorder with both hands, but the next shot shows him holding it in one hand.
Space Seed - S1-E23
Continuity mistake: On the Botany Bay, McCoy is not holding his tricorder, but when the ship's lights activate, McCoy suddenly has his tricorder in his hands.
Space Seed - S1-E23
Continuity mistake: When Khan asks Kirk "How long?" Kirk's head is near Khan's with nobody behind his head, but the next long shot shows Scotty behind Khan.
Space Seed - S1-E23
Continuity mistake: Khan's hands switch positions several times on the table between quick cuts as he and Kirk talk.
Space Seed - S1-E23
Continuity mistake: At the compression chamber, the pressure meter goes from thirty to ten too quickly to be accurate between shots. After showing ten, the meter jumps back to twenty.
Space Seed - S1-E23
Revealing mistake: Just before Kirk and Khan fight in Engineering, Khan bends Kirk's phaser pistol in half, but the small hand phaser on top is still functional.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.