Qualifications
If you can get someone to hire you as an agent, then you're an agent (well done). If they were willing to hire you without any college degree, job experience, or knowledge of the entertainment industry, then you're a really lucky agent who likely has some great family connections.
Most agencies will at the very least expect you to have some sort of experience in movies, TV, commercials, music, books—whatever industry you're trying to represent. Said other requirements depend entirely on the company doing the hiring.
Maybe you don't need a college degree, but it'll certainly boost your chances—especially when you're starting out and you're still building connections. A college that specializes in the entertainment industry or the arts is a great place to build those connections. You'll also have to go through the company's junior agent training program—a.k.a. mail room duty followed by running errands and getting coffee (source).
If you've got a fancy graduate degree or even an MBA, you'll probably have loads of doors open to you. You'll probably still have to work your way up to senior levels, but at least they won't start you in the mail room (you lucky duck).
The qualifications expand if you're trying to open your own agency. They don't let just anyone in Times Square become an agent; there are rules. First thing you'll need is a license to do business as an agent—not necessary in many states but an absolute requirement in California and New York (as well as Texas and Florida), meaning if you want to represent talent who lives in those states you better have a card that entitles you to do so (source).
You'll also end up paying business bonds and fees to operate in a few states. Then there are the business rules of whichever state you live in that you'll have to follow as well.
Should you open up your own agency? You can, but you better look at this as more than just an easy way to get your name in Variety. After all, some people may remember your successes, but everyone will remember your failures.