In many episodes, Eagle pilots fly missions in their usual gray uniforms; very rarely do they wear pressure suits ("spacesuits"), and they almost never wear helmets. Since all of their missions involve flying through vacuum, this is a ridiculous risk to take.
In the pilot episode, the massive explosion on the moon is on the far side, yet the moon is thrown out of earth's orbit. it would be more reasonable to assume that an explosion on the far side would have driven the moon into the earth.
The Moon is shown to be traveling within the Lagrangian points of several planets during the series. Not only would this cause serious issues on those planets, it would likely destroy the Moon itself due to tidal gravity effects.
While the core narrative is for the Moonbase Alpha crew to return to Earth, no one every mentions that their space travel at relative speeds and faster would result in time slow down for them, but continuing at the same rate on Earth. This would mean that while years passed for them decades or even centuries are going by one Earth. Even if they did somehow manage a return, they would do so centuries after they left.
The lack of resources is frequently referred to throughout the series. This is contradicted by the fact that throughout dozens of Eagles are destroyed. As they never run out they must be replacing them, the alternative is they have dozens of spare craft on a moon-base.
The Moon would have be accelerating at an exceptional rate to move from solar system to solar system each episode. How the Eagles (the sub-light speed spacecraft used by the cast) are able to keep up with or return to Moonbase Alpha when they depart is never explained as it would be nearly impossible to do so.
The Alphans don't have the sort of translator technology that other science fiction series use to get around the language problem, and there is seldom any explanation how the Alphans are able to communicate with races from planets thousands of light years away from Earth.
In scenes outside the moonbase, actors move in exaggerated "fake slow motion" ways in an attempt to simulate moving in the moon's low gravity. Even by the standards of the era, they look downright comedic at times.