If you move from your home to enter into a living arrangement with a disabled person and three weeks later they want you to leave can they be sued?

Anybody can be sued for anything, but unless the other person promised to let you live there for a longer period of time, or to give you a certain amount of notice before kicking you out, or agreed beforehand to some sort of limitation on their right to kick you out, I don't think you have a very good case.