The shuttle's original purpose was to act as a reliable, cheap and fast launch system, and although it was superbly awesome, the shuttle failed in it's objectives. Hence, I doubt another shuttle-like system will be built, as recovering first stages SpaceX style or just recovering engines as ULA planned may be cheaper or safer. Answer: Yes, a private company could build one (given money, motivation and time) It's just that another shuttle wouldn't serve much purpose. (Dosen't mean I don't want another shuttle, though.)