You are probably thinking of a 'thin client' and afaik it's not practical for home use. There have been various takes on the idea, but usually involve a small 'device' on the desktop to control keyboard, video & mouse (KVM) and connect to a 'server' (either dedicated or using something like terminal services). Very good for control and management in a corporate environment.
You could potentially get some long KVM cables and put the base units under the stairs, but then you may have cooling issues and then can't get at the CD drives.
A desktop format PC might be more practical for saving space or, if money is no object, one of
these (PC built-in to LCD monitor)