Move water shutoff?

Hi, I'm trying to do research on this and I think my google-fu may be failing me or something because I feel unable to get a sense for how possible my desires are.

Pictures ft. cat and David Bowie: https://imgur.com/a/M0eUY6E

My water shutoff is located in the middle of the only full wall of my basement, the wall where I would naturally put the TV/face the couch were it not for the sweet little closet containing my water shutoff. A previous owner DIY-finished this basement and covered up the issue by making this. Not great, but I get it, and it's not a bad solution.

Is it possible to....make my basement a real room and move the shutoff system somewhere else? Even the corner of the room instead? I'm guessing it's possible, but is it possible without draining all of my savings for 3 feet of wall? Do I even own these pipes, or is the city involved with this? Any advice you can offer, I would really appreciate it. There's already a basement bedroom and I'm planning to add a basement bathroom and it'd be nice to have a more easily usable living area and make this a real space.