Factual error: According to "The Doomsday Machine", full impulse drive is one-quarter the speed of light. In the first two movies, Enterprise used thrusters as opposed to impulse drive to leave Spacedock, confirming the notion that impulse drive is far too fast to leave such a (comparatively) small structure. Styles, however, orders Excelsior to one-quarter impulse, which is 18,750 km/s. In one second, she will travel half again Earth's diameter. From the time he gives the order to the time we see Excelsior clear spacedock's doors is approximately 40 seconds. Even allowing 30 seconds to go from rest to one quarter impulse, spacedock must be 13-15 times bigger than Earth! That's some serious engineering. (00:23:45)
Revealing mistake: As the Enterprise nears the space dock doors there is an outline around the ends of the nacelles that precedes them as it moves. This shows that the ship was superimposed over the shot of the inside of the space dock.
Other mistake: On the backside of the DVD near the bottom is a series of six images purportedly from the movie. The second image shows the crew in uniform, including Mr. Spock, standing in the Federation Council chamber. This image comes from the movie Star Trek IV: The Voyage Home (after the crew has returned to Earth and is waiting to hear the Council's judgment).