1. Set up a tent at a craft show.
2. Put it in a zine and
3. Make friends with a band and do their art.
4. Check out Burning Man.
Artists have always had good reason to avoid galleries. In an ideal world you'd submit your work to the right gallery, they'd do the legwork to sell it, and then take a cut from your profits for their trouble. But most gallery people are business people, and most artists are not. Artists often get screwed.
Some artists simply resent the elitism of galleries or their work just doesn't fit in that format. By the '90s artists had organized cooperative galleries owned and run by the artists they showed. These can be a blessing or a curse, mostly due to the group dynamic involved. Some artists had gained popularity with subcultures like goths, geeks, and bikers.
Then came the Internet. Now most artists have their own website to publicize their work and thus sell it themselves. There are also online galleries, but I haven't heard much about their success. Many young artists feel empowered to start collectives and businesses. Real galleries still matter (a lot) but here are some alternative marketplaces and art spaces I admire and that I hope will change "the system:"
DeviantArt. Yes, deviantart.com has given voice and community to many a young outsider. You used to have to work really hard to find even a single other person in your community/high school who shared your interests, unless maybe you lived in NY, LA or Tokyo. If you don't believe me go watch Ghost World. Nowadays you can get feedback and inspiration online, it's great.
Mail Art Projects. A directory of calls for submissions for mail-in art projects and shows. Remember "Post-Secret?" That kind of thing, often collaborative and anonymous.
These are just some, but I hope to keep a series of posts going about alternatives to galleries, tagged not-a-gallery.