Corrected entry: During the first act it's shown that the rocket has a thick section of sheilding between the crew compartment & the nuclear reactor, to protect them from radiation. But later during the midflight EVA one of the astronauts travels down to the aftward end of the hull, right in the neighborhood of the reactor. The astronaut would've been dangerously unprotected.
Corrected entry: Just after landing on the Moon the crew speak with a radio host on Earth, who says "Mr. Barnes, can you tell us where you landed. The astronomers at Palomar say they could see you if they knew where to point the big eye." This is not true at all. There wasn't a telescope on Earth at that time capable of resolving an object the size of their spaceship at that distance. This is a fact any astronomer would've known even then.