Short answer, yes it's possible. Long answer, no. Just don't.
First, with web servers, the important part of your bandwidth is going to be the upload speed. You'll want at least 5mb for a tiny site, so 370k ain't gonna cut it. Secondly, there's the security (or lack thereof). Thirdly, for the amount of ballache it is to setup a server with the correct security, software etc. and the price that decent hosting can be had for, it's just not worth it doing it yourself.