Factual error: Between Earth and the Moon the crew of the spacecraft have an EVA in which they walk on the ship's metal hull with magnetic boots. At one point an astronaut inadvertently slips both boots off the hull and drifts off into space, helpless. There is a long shot of him drifting away in which his body spontaneously spins around 180 degrees and then stops rotating, which is all a violation of conserved angular momentum.
Revealing mistake: The crew of spaceship Luna make an EVA to dislodge a stuck radar antenna on the ship's hull. One astronaut pokes a tool into a cavity to dig out an interfering mass of machine grease. As he digs, the material comes out of the hole and rolls down the side of the skin of the rocket under its own weight, rather than floating off weightlessly into space.
Revealing mistake: During the rocket launch, Joe needs to raise his hand up to his panel to activate an instrument. The rocket is accelerating at many times Earth gravity at sea level, so it's a very strenuous task for Joe to lift his arm. Nevertheless, as he does so, his shirt sleeve is very lax, hanging loosely, whereas it ought to be draped rather heavily.