Qualifications
The first qualification you need to meet is turning twenty-one years old. Once you've hit that incredibly low bar, you must also have nothing on your record (like a felony conviction, or threats against a public official, or the inability to see) that would prevent you from owning a firearm (source).
The next step on your way to gun store ownership is the permit. To obtain that, you'll have to fill out some forms from the Bureau of Alcohol, Tobacco, and Firearms. You'll also have to tell local police (by mail) that you're opening up a shop, because they need to know who's a legal weapons retailer and who's a weapons trafficker. Believe us, not having a SWAT team show up at your house at 3:00AM makes the cost of stamps totally worth it.
Once you're approved, you'll have thirty days to comply with local regulations, whatever those may be. From that moment on you're a legal firearms dealer, weapons purveyor, and a hero or villain in the eyes of your neighbors.
You didn't see "college degree" in there because you don't actually need one for this job (source). If you're going to work at someone else's store, they usually just want to know you can sell guns and properly use them. If you're opening your own store, then you probably already think you're qualified enough.
That doesn't mean college degrees won't come in handy; the knowledge you get for taking business classes will play into how you run your shop, and history degrees can be useful for the antique gun specialist. Just remember, that no matter what your area of study is...you probably still can't bring guns (even antique ones) to class.