Possible, probably. Look at LINUX LiveCD distributions, you'll probably be looking at customizing one for the remote control usage and installation of Flash Player, Quicktime, Mplayer, VLC, so on. You'd be out of luck if broadcasts used most kinds of Windows DRM or active X stuff. It would need to have persistent storage for bookmarks to the station sites et. al.
Mythbuntu users might have some tips on how to do it; I don't think Mythbuntu itself will do it by default though, not with all the codecs you'd need and the remote control and USB boot.