Qualifications
First and foremost is that voice of yours. Whether you have the deepest basso profundo or can reach frequencies that only dogs can hear (or you're stuck intermezzo, which is totally fine too), your entire career depends on your abilities on the stage. You're going to have to be able to do more than carry a tune. That trill sound you hear opera singers use is called vibrato, and it comes with years of training and experience.
The acoustics of the opera house are made to carry your beautiful gift to the rafters, but there's still got to be power behind it―not to mention knowing when to pull it back for effect. This is why opera singers spend years practicing, rehearsing, and honing their craft―they even learn the best times and ways to breathe.
A degree isn't necessarily necessary, but it will help open doors. Directors and producers will be looking to hire the best, and if your résumé lacks that BFA, MFA, or conservatory work, you better wow them with your gifts and show them how professionally you can carry yourself (it's a chin up/nose down kind of thing).
And don't forget about the side work: singing is only the tip of the harmonic iceberg. Opera is a performance art, and you'll often find yourself doing more than just standing there. Movement is important, and depending on the show you might even have to dance.
Acting skills are also useful; the music and words can only carry meaning so far, it's up to your delivery and connection to the music and words that will hit the audience right in their feelzones. A variety of performance knowledge is good, because it tells people how totally opera seria you are about your abilities and reputation.
Finally, you must not object to wearing heavy costumes, ridiculous wigs, and layers of makeup under hot stage lights. You'll spend most of your time feeling like you're melting. That's show biz for you.