Crew or equipment visible: When Ford and Arthur meet Zaphod for the first time in the series (near the end of the first episode) the operator of Zaphod's third arm can be seen (they can be seen in other episodes, too, notably the third)
Continuity: In the first episode, when Ford and Arthur are in the pub, there is an outside shot of the pub. The pub is illuminated by a low evening sun; whereas in other outside shots, and on the pub clock, it is clearly nearer mid-day with bright sunshine.
Errors made by characters (possibly deliberate errors by the filmmakers): In episode 4 computer Deep Thought is speaking about building the computer Earth, and he calls it "the computer who will give the answer to the ultimate question". Actually the sentence should be the other way round: "the question to the ultimate answer" - they already have the answer (42).
Continuity: In the second episode, when the Vogon captain is reading his poetry to Arthur and Ford, Ford's hairstyle changes noticeably between the first part of the scene and when it returns to them after the Guide's explanation of the worst poets in the Universe; the result of the two parts of the scene being shot five months apart. Of course, it could be that the poetry is so horrible that it made Ford's hair curl.
Revealing mistakes: Episode 2: When Arthur and Ford are following Marvin through a white hallway, where Marvin mentions that the ship has been stolen, when Marvin walks toward the camera to enter a doorway, the actor's nose can be seen through the mesh of his mouth.
Continuity: When Zaphod Beeblebrox is at the controls of the Heart of Gold, snowflakes appear in his hair. There was supposed to be a scene in which Trillian creates a snowstorm while Zaphod is sunbathing. That scene was filmed, but was cut from broadcast. This is the only known deleted scene from the series.