Odds of Getting In
Anyone can become a beekeeper. In fact, urban beekeeping has experienced somewhat of a resurgence in the past decade or so, and more and more cities are agreeing that it's okay for people to keep hives in their backyards (though not all; it's still illegal in some cities).
But if it's becoming a "professional" beekeeper that you're after—you know, someone who works for a company or starts up his or her own company—well that's a bee of a different color.
A proficiency in entomology (the study of insects) will help your chances, as will your willingness to pursue internships in the field (paid or otherwise). Internships are sometimes formal, sometimes not so formal, and can be found anywhere from a professional beekeeper's backyard to the site of a major corporate honey farm.
Most of all, as with many professions, making yourself and your interests known is key. That means conferences, networking, and maybe a midnight retro screening or two of Jerry Seinfeld's Bee Movie.