I think it means to have a lot of hair on body! It's literal, not a slang.

"Things began to get hairy" = the situation began to look uneasy/difficult, or even dangerous. This explanation is appropriate, This phrase from the movie. Boss praises the killer, said that only he can be trusted with the most complicated deals.

very slang...

"Things began to get hairy" = the situation began to look uneasy/difficult, or even dangerous.

What is the context? because it does have a slang meaning, usually used in extreme sports.