Deliberate mistake: For reasons unknown and unexplainable, the otherwise spacious and well-equipped Moonbase has halls and doorways too short for the average human to fit through without ducking. (00:20:00)
Other mistake: Lunar Module 32 has number 4614 and more on its side; then it explodes upon re-entry to Earth. But later, when Paul takes up Lunar Module 29, it has the same numbers and markings as LM 32. (00:15:00 - 00:27:10)
Continuity mistake: When the Apollo 8 space debris is shown, the print type of "USA" on its side changes between the long shots and closeups. (00:04:20)
