I hate my cat, how can I get rid of it?

The cat on my ship annoys me, as it keeps following me and, when it does, it parks itself in front of what I want to interact with, thus preventing me from doing so unless I move the mouse away from it.

Something like this is a pain with the shiplocker, because here it sits and covers most of the shiplocker and there's only a tiny bit I can check before my mouse is off it. More annoyingly, it sometimes parks itself at the flight control and I can't even access it.

How can I get rid of my cat?

Without modding, it's impossible.

However, with the Purchasable Pets mod, default pets (the one that spawned with your ship, depending on your race) are removed from the game when you start a new character. Instead, they can be purchased from the Infinity Express at the Outpost.

You can also remove existing pets with the Pet Trap item it (the mod) introduces.