Probably the best way to go about doing this is configuring the network so that each computer has a static IP that cannot be changed by the normal user-level accounts and implementing a firewall that only allows IP traffic to be forwarded to/from the internet for specific IP addresses on the LAN. You can purchse a router/firewall that can do this, of if you have an old computer laying around and the proper knowledge, you can put 2 ethernet cards in it, install Linux, and set it up to be a router/firewall.