What does 'geek out' mean?
In Looking

1.

(Patrick and his co-worker Owen attend a video-game launch party.)

Patrick: You know what? I'm gonna put the whole Richie thing behind me.
I'm just gonna have fun and geek out.

Owen: We should get totally drunk in this.

What does 'geek out' mean in this context?
Does it mean 'to have a really good time'?

2.

Agustin: I mean, he owned what he does.

Patrick: Oh, so you want to be a prostitute now?
It doesn't seem like a very smart career move.

Agustin: It'd be nice to make some money though.

What does 'he owned what he does' mean in this context?

I'd really appreciate If anyone can help me out with these two questions.

Question

What does 'geek out' mean?
In Looking

1.

(Patrick and his co-worker Owen attend a video-game launch party.)

Patrick: You know what? I'm gonna put the whole Richie thing behind me.
I'm just gonna have fun and geek out.

Owen: We should get totally drunk in this.

What does 'geek out' mean in this context?
Does it mean 'to have a really good time'?

2.

Agustin: I mean, he owned what he does.

Patrick: Oh, so you want to be a prostitute now?
It doesn't seem like a very smart career move.

Agustin: It'd be nice to make some money though.

What does 'he owned what he does' mean in this context?

I'd really appreciate If anyone can help me out with these two questions.

Answer

1) "to geek-out" is an idiom (숙어) which means "to act like a geek". A "geek" is someone who is really into something, many times it is games and video games and computers, but now it can also be used if you are really into music or movies or whatever: music geek, movie geek, etc.

2) "to own what do" is also an idiom. It means "to be really good at what you do". Simple as that.