Qualifications
First off, it's possible to find employment as a web designer with no more than a high school diploma or an associate's degree in web design. What's important in this career is experience: Are you an HTML pro? Do you know your Photoshop, Illustrator, and Flash? Can you make pretty, pretty websites that function the way they're supposed to?
That said, if you want the option of being able to find good-paying work in places with competitive tech job markets, then you should just suck it up and slog through four years of college. It isn't like you have to go to MIT for your bachelor's degree; a state school will work fine. Also, while some employers will want you to have majored in computer science, degrees in graphic design, visual arts, or visual design are common and perfectly acceptable in the industry (source).
To succeed as a web designer, you'll need to be creative (of course). You'll have to be detail-oriented; one false move in your HTML code, and whatever website you're working on will look super screwy. Whether you're working for a large company, a small non-profit, or freelance, you'll need to be able to interact with others, be they co-workers or customers.
Finally, given that fully one-quarter of web designers are self-employed, you'll need to be business savvy, which means knowing how to connect to potential clients, being able to manage your schedule and budget, and going through the clustermess that is filing self-employment taxes.