Odds of Getting In
Compared to other centuries or millennia-old fields, nuclear science is a relatively new discipline. We're learning more and more about its potential—in energy, medicine, and other useful applications—as time goes by.
That being said, more and more people are going into this field; more and more people means more and more competition. It helps to specialize, which means getting extra training. Know what you want and work until you get it—it's what your nuclear researching forebears did.