Hmm, I don't know how exactly but I think it's possible. I've read something about using a HDMI Cross Switch to send the signals across the house but I don't know how to explain it to you because I've never done it in my life. All I can tell you for sure is that it can be done.
I think you can with the controllers but you are talking about the entire system, correct? I doubt I would attempt it myself, for some reason the house we live in hates anything wifi related. Even trying to use a tablet sometimes between the rooms can be difficult.
You could try using really long cables if the wireless method doesn't work. My console is in another room and I actually have a cable that is long enough to reach the room I play in. It's not perfect but improvisation sometimes works.