This article is from the Ballet and Modern Dance FAQ, by Tom Parsons firstname.lastname@example.org with numerous contributions by others.
It's not impossible--it has been done before--but the odds are
against it. Leigh Witchel offers more details:
The average age of a woman starting ballet is between eight and eleven,
of a man, often in his teens. Later is not unheard of; Melissa Hayden
began at 15, Igor Youskevitch at 22. A word of warning, however: As
you grow older, developing flexibility is infinitely more difficult.
If you do not have a natural facility, you will be fighting an uphill
battle the whole way and may find the pain too great to be worth it.
Also, for a woman, developing the ankle strength and articulation of
the feet necessary for pointe work takes around five years, which adds
another handicap. Moreover, at the onset of training, you can really
only take so many classes a week without risking injury. So the roads
of an amateur and professional do not diverge until at least a little
way into training. At that point, take a good look at what you are
doing, your progress in relation to others, and how happy it is making
you. Are you ready to play a game of catch-up that may be sisyphean?
It may be worth the struggle.
See also the following question.