Frankly, start with a taser and some pepper spray. If you're really concerned, a bulletproof vest is probably a lot more helpful against threat of gun violence than yet another gun.
Short answer: it is probably possible to acquire a firearm, if a person were comfortable with the potential (extremely serious) consequences. Note: I said "acquire." I did not say "legally purchase and own," much less "legally possess."
...except, of course, in Texas.
(^_^ bet nobody saw that coming.)
Legally, at least in Texas, all you need is a state-issued ID to purchase a shotgun (not 100% sure about handguns.) There is no waiting period for shotguns and no state registration. There's probably an exception expressly to permit hunting tourism. (By the way, hunting requires a license. Otherwise, you're poaching.)
point: unless you're actually here for hunting tourism, forget it. Srsly.
American rights only extend to Americans. Non-citizens are protected by law from crime, but don't qualify for stuff like the 2nd Amendment - which, btw, absolutely does not automatically confer the right to carry a weapon, concealed or open. Beyond private property, very few people have a concealed carry permit and open carry in public is very rarely legal outside a handful of specific situations (for example, transporting.)
The only exception I can think of would have to be grandfathered (what qualifies varies depending on location, type of firearm, etc.) "Grandfathered" firearms generally refers to antiques or historical reproductions - ie, black powder, muzzle-loading, etc. - as exempt from many of the usual restrictions on purchasing, possession, etc. (Some kinds of guns are always illegal for non-military / law enforcement. Generally, it's safe to presume that this applies to any non-grandfathered, functional firearm that takes a bayonet or needs wheels.) In this context, a reproduction revolver, such as a Civil War era Colt .45 or Remington .44 might be completely legal - especially if you ordered it as a kit. Granted, it's a six-shot revolver that's almost a foot long and takes fifteen or twenty minutes to reload, but that would probably be the most realistic, potentially legal possibility.
Which leads to the question: ...what would you DO with it? When Americans discuss "self-defense" in relation to firearm, we pretty much mean against intruders specifically of a private residence.
As a rule of thumb, people in America do not walk around packing heat.
Concealed-carry permits are pretty rare. Having a permit doesn't make it legal to bring a gun any old place - ie, it is always a felony anywhere liquor is sold or gambling takes place, ao gas stations, bars, liquor stores, casinos, many restaurants, department and grocery stores, also within a thousand feet of a school, government bldg, airport, hospital, pharmacy, etc... Open-carrying a firearm, whether or not it's technically allowed, is absolutely guaranteed to freak people out. Americans tend to presume a gun is loaded (whether or not it actually is, they will behave as if it is) and will probably call the cops, attempt to negotiate, etc. as if a crime is in progress. Life will get awkward, and only get more awkward when the cops arrive. Law enforcement tends to react very, very poorly to any situation even potentially involving a firearm, especially a firearm they can't see. Hence the terms "armed and dangerous" and "suicide by cop."