Let me give you a very brief explanation of United States firearm laws (on a federal level, not by state):
1) If you are a felon, have been dishonorably discharged from the military, are currently under indictment for a serious crime, have been convicted of domestic violence, etc., you are not legally allowed to own or possess a firearm. Your rights may be restored in certain circumstances. A person who has been involuntarily committed or has been adjudicated as mentally unsound is typically also prohibited from owning a firearm.
2) You must be 18 to purchase a long gun (i.e. shotgun or rifle) from a licensed firearms dealer. You must be 21 to purchase a handgun from a licensed firearms dealer. Private party sales are regulated by the state, but generally, you must be 18 to purchase either a long gun or handgun in a private party sale. Some states do prohibit possession of handguns by anyone under 21 years of age, which would preclude private party sales of handguns to persons under 21.
3) Fully automatic firearms, guns that would typically be considered long guns but have short barrels, and "destructive devices" such as explosive devices and very large bore guns are regulated by the Bureau of Alcohol, Tobacco, Firearms, and Explosives (BATFE). To own one of these, a person must otherwise meet all requirements of purchasing a firearm, be 21 years of age, and pay a $200 tax stamp to the BATFE. Their application to purchase said firearm may take 9 months to process, but if they are approved, they can purchase these devices. Their tax stamp MUST be with the firearm or destructive device at all times. (Special note: that's the EXTREMELY condensed version of the 1934 National Firearms Act.)
Now, I tell you all this for a reason. Consider what the implications of these laws are. First off, criminals cannot legally own guns. Gang bangers, murderers, rapists... They can't own guns, by law. Does that stop them? Obviously not.
So the extant laws have done nothing to really stop gun crime. What they have done, however, is to limit LAW-ABIDING CITIZENS ability to access firearms. Should we make laws stricter? Well, ask yourself this: why? Why, when criminals will still have access to guns, should we make laws that really only affect law-abiding citizens more strict?
And to answer the question in your title:
Yes, people should be able to buy guns. We are endowed by our Creator with certain rights- life, liberty, property. We have a right to live, we have a right to be free from tyranny, and we have a right to own and protect our property. Without those three things, the rest of society is meaningless.
Guns, when originally introduced in Europe, challenged the status quo. Initially, a soldier had to train for most of their life to be anywhere near the level of a mounted knight, and to own a horse, sword, lance, and armor was expensive. A peasant archer could not go toe-to-toe with a knight and expect to win. However, with a gun, the peasant could defeat a knight's armor, meaning that a dishonest nobleman could not deprive the peasant of his life, liberty, or property. Guns did not require nearly the amount of training as a longbow or a sword. They put power into the hands of the general populace.
Which is why gun control was enacted in England from the 1600s on. James II made it illegal for Protestants to own guns, because he was a Catholic and did not want to lose the seat of power he possessed. Later Acts of Parliament reinstated those rights.
Note that every time in recorded history where government edict gave government a monopoly of force, there was tyranny.
Now you answer the question: should people be able to own firearms?
EDIT: @Space Cowboy, I think this is a different "A."