Qualifications
First and foremost—fearlessness. A sense of invincibility. A delicate balance of arrogance and ignorance. Being young helps, too. Because if you don't feel like you can do anything, you won't succeed here.
Second—responsibility. These two may seem at odds with each other, and to a certain extent they are. But you can't be entirely reckless, even as you're doing things most people wouldn't do for love or money. There must be a precision to your actions so you don't wind up killing or severely injuring yourself. Or others. But mainly yourself.
This is where training plays a huge part. Before being accepted on the Equity Stunt Register (which you need to do if you're going to make a living doing this), you'll need to become extremely proficient in a number of areas, including different forms of fighting (judo, boxing, hair-pulling, etc.), riding, underwater stunts, and more.
You'll want to be a black belt in at least one of the martial arts, and excel at just about every gymnastic endeavor you try your hand at. In fact, many stunt performers are former circus performers. So maybe go join a circus. Then, when you're done clowning around, follow your calling in LaLa Land.
Lastly, you're still acting. You may need to embody the same character that someone else in the movie has, and do so convincingly. For that, you should also have some theater training. You might not be able to make a living solely on your acting alone, but it'll be as helpful a skill to have as holding your breath for over a minute.