A very interesting thread over at rec.guns.

The best answer to the question is by Mark Crispin

# Is there a legal way for me to buy guns for self-defense and/or
# recreational plinking?

The US government has a long series of FAQs on this question here:
http://www.atf.gov/firearms/faq/faq2.htm#r

Short answer: probably no.

Longer answer:

Non-immigrant aliens (which includes tourists) in the USA are generally
prohibited from possessing firearms.
With the proper US export paperwork and home country import paperwork, you
may purchase firearms in the USA. However, the firearms must be shipped
by the seller to your home country; you can not accept delivery of your
firearms in the USA.

There are certain exceptions to this prohibition. The easiest way for a
non-immigrant alien to legally possess firearms in the USA is to obtain a
hunting license from any state. This license must not be allowed to
expire, as it effectively serves as your alien firearms possession
license.

There are also state requirements. For example, Washington State has no
licensing requirement for citizens to possess firearms, but does require
aliens to obtain a Washington State alien firearm license. Only two
states (Alaska and Vermont) permit carrying a concealed pistol without a
license.

Even after you obtain a state hunting license, you won’t be able to
purchase and take possession of a firearm in the USA due to the transient
nature of your vis
it. The ATF web page above has detail about the
requirements.
However, with your hunting license, you can borrow someone’s firearms for
plinking, hunting, etc. You probably will not be able to get a license
to carry a concealed handgun.

I’m sorry that we’re so unfriendly to visits these days. Blame cretins
who decided to take out their rage against the world by coming to the USA
and killing Americans.

One of the most notorious of these incidents happened on the Long Island
Rail Road in New York State in 1993. A Jamaican lunatic named Colin
Fergunson killed six people and wounded nineteen others, claiming that it
was justified by “Black rage” against his White and Asian victims. In the
aftermath we gained one of the most anti-gun members of Congress.

Read the whole discussion here.