If you're just running a web site without much dynamic content over a home-sized connection, you can get away with a very low-powered computer and still be fine. For example, I run my (admittedly very low traffic) web site on a Pentium 166 and it works just fine. (Although I'm using OpenBSD instead of Linux, there shouldn't be a huge difference in requirements.) So you could conceivably get yourself a very cheap computer to run your web site as long as you don't want to have big heavyweight applications running on it, like GUI frontends and databases and office suites and modern browsers. Personally, I think that you'd be better served by learning Unix command-line stuff rather than arguing with GUI nonsense first anyway, assuming that there's a Linux distribution these days that supports not having GUIs installed.
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Bitt Faulk