The Second Amendment of the U.S. Constitution explicitly states a private citizen's right to own and bear arms.
Amendment II: A well regulated Militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed.
Until such time as the Second Amendment is repealed or amended itself, that is the law of the land. There should be no problem under existing law for legal, non-criminal, mentally fit individuals in applying for a license and owning a gun.
My subjective opinion: I think responsible citizens who legally own guns are a very good crime deterrent; I think legal gun owners are more likely to be trained and knowledgeable about gun safety; I think (I'd have to do more research) that the majority of gun crimes are committed by outright criminals who would pay no attention to an anti-gun law. I'm no law scholar, but I feel that passing anti-gun laws underneath the Constitution is an attempt to avoid honestly discussing the issue.