Let's say that you have telnet access to this machine with a login shell. Then you can easily obtain open source web server software, install it, run it, and set it up to execute CGI's.

This would be circumventing the intent of your sys admin for the web box.

On a more helpful note, you can easily set up any computer to be a web server. Any old box will do, virtually any OS, although a Linux install would be a nature fit here. Taking this route you may circuvent objections rather easily.