Qualifications
All you have to do is pray for it.
Kidding. That's what you'll do for the rest of your life, but for the meantime you need to do some soul-searching—and a lot of paperwork.
To be a nun or a monk, you need to actually feel that heavenly spirit flowing through you. This isn't the place for perpetually ambivalent, hyper-intellectual whatdoesitallmean-ists. This is a place for conscientious believers, free of doubt. If you're the type who starts a lot of sentences with the word but, then this definitely isn't your ideal work environment.
There's also the charitable side of the gig. It's not all silent contemplation; nuns and monks are out helping people in need all day, every day. Folks of the cloth take the "there's no I in team" mentality to a whole new level.
Think you have all that? Great. Now you have to do what you do when applying for any job or college. You'll need to supply a complete work history, legal history, job history, financial report, a couple letters of recommendation, and go through an interview process of some sort. They may be sticking you in a small cell, but they want to make sure there isn't a different kind of cell waiting for you in another state.