It can be done but its not exactly easy.
Windows Server has a proxy service which most recently was called Fore Front Threat Management Gateway but I believe has been removed completely now from support.
Anyway there are two ways, to run a 'true' web proxy you can setup the proxy on that spare computer, plug in to the network on a static IP and change all users proxy settings in internet options to hit that hostname/IP. The second way is to use it more as a firewall by running it similarly to how I do in my lab which is install two NIC's on that spare computer, run one to the router and one to the network (switch) change your DHCP gateway address to use the IP address of the network (switch) NIC. This way all traffic runs through the spare computer and the free firewall tool you have on it, this way you can control bandwidth and limits, etc.
Have a look at freeproxy for first option and pfsense for second.