You can actually can use that ip range on your local area network. However, keep in mind that traffic destined to a target IP in that range will remain on your network. For example if you wanted to access an Internet web page that resolved to an ip in that range, the traffic would not leave your network. So it isn't recommended you use public routable IPs on your network unless they have been assigned to you. There are still many universities that use public IPs within their network...they just firewall themselves from the Internet of course.
Again, computers are not programmed with the concept of public/private IPs so any ip range will work. The problem is that if you use a range that is not assigned to you, you are going to run into issues is you need to get traffic across the Internet to that address. So in summary use a private up scheme on your local network.