In 1999, humans built a space station in the Mmoon where they stored all nuclear waste from the Earth.

"in the moon" only works if it was under the surface, I have not seen the show so cannot tell if this is right.

This should be preceded by something like "in the show...".

The explosion was so big it made the Mmoon escape Earth's gravitational field and travel into deep space of the universe.

With each episode of the show, the Mmoon travels further and further from the Earth, briefly stopping in star systems along the way.

In 1999, humans built a space station in the Moon where they stored all nuclear waste from the Earth.¶
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Dictionary dot com has a nice short article called "when to capitalize earth" that talks about the differences between Earth, the earth, earth, etc

Space: 1999 is a British sci-fi TV show from the 70's that I watched as a kid.

Very minor corrections, 70's is more correct, but it isn't worth worrying about

In 1999, humans built a space station ion the Moon where they stored all nuclear waste from the Earth.

"the Earth" *could* work, but sounds a bit awkward here. If you're treating Earth as some sort of separate place, like as if it were a country, try to avoid using "the Earth."

The explosion was so big it made the Moon escape Earth's gravitational field and travel into deep space of the universe.

WithIn each episode of the show, the Moon travels further and further from the Earth, briefly stopping in star systems along the way.

As the Moon stops, the inhabitants of the space station visit the nearest planet and encounterhave some adventure.

Encounter isn't explicitly wrong here, but it doe

I watched it so long ago that I don't even remember if they managed to go back to the Earth or not.

Space: 1999 is a British sci-fi TV show from the 70s that I used to watched as a kid.

In 1999, humans built a space station ion the Moon where they stored all the nuclear waste from the Earth.

One day somea thing happened, thate thing which led to a nuclear explosion.

The explosion was so big it made the Moon escape the Earth's gravitational field and travel into deep space of the universe.

As the Moon stops, the inhabitants of the space station visit the nearest planet and encounter somehave an adventure.

In 1999, humans built a space station ion the Mmoon where they stored all nuclear waste from the Earth.

The explosion was so big it made the Mmoon escape Earth's gravitational field and travel into the deep space of the universe.

With each episode of the show, the Mmoon travels further and further from the Earth, briefly stopping in star systems along the way.

As the Mmoon stops, the inhabitants of the space station visit the nearest planet and encounter some adventure.

I watched it so long ago that I don't even remember if they managed to go back to the Earth or not.