I decided at twenty-four to make dancing my career. It has proven feasible though is still not independently economically viable. The thing is to ask what you mean by a career dancer. I regard dancing as my career because it is something I'm dedicating my life to, working 40+ hours a week at, and generating most of my wealth from. In my view that makes me a career dancer.
On the other hand, some people have a lot more in mind when they think of a career dancer. If you mean a professional competitor and a champion at that, well there's Tony Dovolani and Jesse Desoto as ept777 mentioned, but other than them I can't think of anyone who started beyond their preteens. The thing is, though, most of us don't have the talent or drive to reach those heights, and for us that's okay. Our goal is simply to dance as much as we can.
Ask yourself why you're doing this. If you haven't taken dance lessons before how do you know you love dancing so much, to the point that you're ready to devote your life to it? Have you thought about those days when you're bone-tired from staying up til 2 AM getting your open telemark right and you still have to come into the studio to teach three classes and four private lessons without a break? Have you considered how you'll keep your smile when your students are not the friendly, exuberant learning sponges you envision but ordinary people, many of whom are only there under duress? How much have you thought about the fact that your real job will not be a dancer or even a teacher but a salesperson and that everything else, including the dancing and teaching, is just part of your sales technique? What about never having a life outside dancing because your studio hours are deliberately scheduled over most people's free time?
If you can unhesitatingly answer, "Yes!" then I say go for it. Go for it and don't let go. If your answer is anything else, I say go for it anyway. Dancing is a wonderful thing to do with your life, and if there are more obstacles than you realized, that's true for any path. Learn to dance anyway and work towards becoming a professional dancer - it's always easier to change your mind and not become one - but keep your eyes open and talk to everyone you can, especially your teachers, about the path to becoming a professional.
Good luck!