5' is not very deep. Perhaps you'd be better thinking about making a platform for it. You should probably NOT attempt to use the closets as a means to support ANY weight. You don't know how they're built inside. It's very UNLIKELY they'd have enough strength to properly support any additional load. Remember, there's more to structure than just vertical weight. You have to take into account lateral motion too. So your platform up there would have to also be securely anchored into the wall as well. Then there's those pipes up there, they look like they'd end up being way too low.
Are you renting this or is it a condo?