I'm pretty sure that Linux will run on a 386 w/ 2MB of RAM. Just don't try recompiling the kernel. RedHat reccomends 16MB of RAM for decent performance, and about a gig of space depending on what you install.
Once upon a time, I had RedHat 4.x on my 486 DX2/66 w/ 8MB RAM. Worked just fine for me. You may not be able to do all the pretty X windows stuff, but you should be able to get things configured in text mode no problemo.
You'll really only need a monitor for initial installation. Once you are able to telnet to the box, you won't need a monitor anymore. I have a linux box sitting here that hasn't had a monitor connected to it in over a year. I just telnet in a few times a week to look at the logs and make sure everything is hunky dory.