In order to become an illustrator, it'd be a good idea to get a fine arts degree, but for many jobs the qualifications simply depend on how good you are, how impressed a potential employer is by your portfolio, and how well your talents fit a client's particular vision.
It's important to recognize that we've entered a digital age, and taking classes in graphic design and other computer-related courses is going to greatly improve your chances out there. That's especially true if you have a degree in one of them. We know it sounds weird that a glorified cartoonist should get a four-year degree. But show any employer a four-year degree, then show them a drawing of a cat, and see which one actually gets you hired.