This page is a set of terse instructions for setting up IP masquerading. It's not very complete but will get you up and running in no time.
I am using Redhat 6.0, so these instructions are heavily geared towards that setup:
A stock Redhat 6.0 kernel will have everything you need compiled in, if you compile the kernel yourself then you need all the firewall and masquerading options enabled.
Let's assume that your internal network IP addresses are 192.168.0.x, add these commands to your /etc/rc.d/rc.local:
export PATH=$PATH:/sbin # Load masquerading modules modprobe ip_masq_ftp modprobe ip_masq_irc modprobe ip_masq_raudio modprobe ip_masq_quake modprobe ip_masq_user # Build IP chain rules ipchains -F ipchains -P forward DENY ipchains -A forward -j MASQ -s 192.168.0.0/24
That's it!
It's magic, here is a pretty diagram, I will explain later on :)

Computers B, C, and D are completely unaware that they don't have "real" IP addresses. As far as they are concerned, they are connected directly to the Internet. When they request something from the Internet:
Yes, that is an extreme over simplification of what happens, but it is all you need to know for now. Some services (like FTP, IRC, etc.) need special treatment, and that is why we load special modules (ip_masq_ftp, ip_masq_irc, etc.) to handle them.
Here are some resources that will give you more information:
Yes, if you really can't find an old 486 to run Linux, take a look at these packages: