Deliberate mistake: Special effects of the ship in space were very expensive in the 60s and couldn't be wasted. So when a larger model was built with slightly different nacelles, shots of both versions became common, even within the same episode. This is why the Enterprise sometimes had red needle-tipped nacelles and sometimes lighted "spinning" ones, and in aft views she had either round white spheres or perforated vents at the nacelles' ends.
Revealing mistake: While the interior sets for the shuttle craft provided adequate room for the actors to stand, the mock-up built for exterior shots was undersized, and too short to stand up in by several inches. So when entering or exiting the shuttle, the actors always had to duck.
Factual error: In several episodes, when the Enterprise has to turn "hard about", we see it banking to one side like a jet plane. The Enterprise has no wings, and they're in space. No banking.
Visible crew/equipment: As the camera looks at the Galileo, if you look at the green background, you can see the top of the sound stage.
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.