Comment by Astrohacker

Public services would be provided by businesses like everything else. "Law and order" would be maintained primarily the way it is now, which is through social custom. But since there would still be aggressors, you would have to pay for private security. And you would probably want to carry a weapon with you.

All of these issues are discussed extensively in the libertarian literature. Murray Rothbard and Stefan Molyneux are good places to start.