It is an adjective and it means "to be reserved". It is used to describe a person who is familiar to you but who tends to keep their thoughts and feelings to themselves.
Ken is complaining about Leo keeping a secret.
- Ken : Hey, why didn't you tell me about your marriage? You're treating me like a total stranger.
おいっ、なんで結婚した事を話してくなかったんだよ？水臭いよ。 - Leo : I didn't mean to keep it from you. Since your mother passed away, you must've had a tough time. I was going to tell you when you settled down.
秘密にしていた訳じゃないよ。お母さんが亡くなって、大変だと思ったから。落ち着いたら、話そうと思っていたんだ。
Meg is worried about Amy because she doesn't look well.
- Meg : Are you okay?
大丈夫？ - Amy : I'm okay...um...it's not a big deal.
何でもないよ...んー...大した事じゃないよ。 - Meg : What're friends for? You can tell me anything.
水臭いじゃん？何でも話してよ。 - Amy : I've been thinking of quitting my job. I've repeated the same mistakes. I'm not good at this job.
仕事を辞めようかと思って。同じミスを何度もしちゃう。向いていない。