2069AD. When her step-mother throws out the stray cat she?s been harbouring, a lonely girl in a sterile biosphere, must brave an unknown, toxic outside world to find her only friend.

I think I agree with Gaenhart. The logline is pretty good.

I don't agree with moses99 at all.

Your suggested logline is pretty dang good. (Maybe remove the comma in front of the word 'must'.)
A couple of questions:? is the date really?necessary, and is there a better word?to replace?"find" that suggests the reason to find the cat other than it being an only friend? It just feels as if there's a?inner journey there that might be worth bringing forward.

How is the cat not dead if it's hazardous? And put the protagonist at the head of the logline;

"A lonely girl must venture outside after..."

How is the cat not dead if it's hazardous? And put the protagonist at the head of the logline;

"A lonely girl must venture outside after..."