Corrected entry: The part of the Lunar Module that Fry and Leela go inside shouldn't be there, the astronauts would of used it to get back to the CSM orbiting the moon. The whole thing lands on the moon but the cabin, the bit where they go in, should actually be floating in space somewhere after they had jettisoned it.
Corrected entry: As shown in other episodes, there are quite a few people who were cryogenically frozen for hundreds of years. If this is so, Then how is it possible they don't know what really happened in the lunar landing? Many more people besides Fry must have noticed something was amiss about the "fungineers" take on the moon landing.