You didn't say where you were located, but if you're in the Annapolis / Baltimore area, shoot me an email. My info is in my profile.
I assume you have some sort of internet since you're posting this. If hosting costs are an obstacle, shoot me an email and I can cut you off a piece of a Linode or something.
As for 'proving you can code', get a free Github repository and start filling it up. Find a project, any project, in any language, and build it. Projects don't have to be amazingly complicated or fit a broad needs. One of my first 'open source' projects was a library that would reliably fetch a favicon from a website.
No, it hasn't made me famous, and no, it hasn't made me rich, but it has helped a few people based on the messages I've gotten through Github. I have had offers from work based on it (not full time jobs, but implementation / freelance stuff), and if anybody questions whether or not I can write a line of code, I can point them at that.
After you write one library / module / program / app / whatever, write another one. The idea being that you'll establish a 'body of work'.
We recently tried to recruit another HNer to fill in on some project work recently and, when my project manager was asking how qualified he was, I just pointed him to his github profile. Not only has he released a lot of code, but he's worked on a lot of the libraries that we're using in our code as well.
Don't worry about whether or not the code you release is 'production ready', or if it's too small, or if you think it's only of limited utility. My favicon parser is under 100 lines of code. Backbone.js is only ~700 lines of code. Code doesn't need to be huge to make an impact. Hell, code doesn't even have to be GOOD to make an impact, but that's a different story.