There are a number of different versions of
libertarianism. Some
libertarians, such as
Ayn Rand, believe that
government has no legitimate role except to prevent the use of force, internal or external. Thus, it could fund a police force to prevent citizens from using force against each other (including indirect force, such as taking people's property even if you can do it when they're not home). A military would also be permitted to
prevent the use of force corporately by other nations. Among the libertarians who agree thus far, there is disagrement as to whether protection of intellectual property from theft is permitted or a use of unprovoked force by government to protect alleged imaginary rights.
Other libertarians reject any form of government at all. Not all believers in this call themselves libertarians, but some do. Others call themselves anarco-libertarians.
The latter idea is simpler to discuss. Suppose all policemen, soldiers, and politicans and employees of state and local governement vanished tomorrow. Various gangs of drug dealers could battle for territory unchecked for awhile, looting and pillaging to their hearts content. Eventually a system of alliances would develop - and eventually it would be discovered it pays to protect people who do useful work, provided they pay taxes to the gangs. This is not so different from the beginnings of feudalism in some times and places. I personally don't like wines, but I am told wines, like governments, mellow somewhat with age. It may be just me, but I've never heard an effective answer to the argument that the primary purpose of government is to keep violent gangs of would be governments down.
The answer to the first argument is more complex, and some mainstream republicans are not far from this form of libertarianism. Is it much more expensive, on a per capita basis, to educate four children than send one of them to prison because he had no legitimate job skills? I've never heard a libertarian discussion of defense pork either. If we do not protect those who cannot afford it from disease, the plague may spread even to those who can afford health care.
On the other hand, I have heard many interesting explanations as to how money and utilities were done better without government, and some of them even seem to have merit. Nevertheless, there are states today with no true government, only fighting gangs of rival warlonds, and they do not seem to have become libertarian paradises.