Piezoe,
You are not a would be libertarian if you believe we live in a perfect world. The idea isn't that there's no theft and bad people around. The idea is that giving the government the monopoly in violence is what gets one the worst deal, where choices are limited and one cannot escape voluntarily using their services.
Private security firms can handle the protection business. Malls and societies can come together to voluntarily pay for competing firms to provide security. Unlike the government provided security, now there's a competition between security firms. Not only do they have to provide security, but have to get it done with least force, as the using excessive force can have a negative impact on their reputation if the competitor is doing it better.
The money that was given to taxes is more efficiently and voluntarily given to firms and insurance companies instead that have better reputation.
Insurance firms become a key in this environment where to get insurance for property, they raise premiums if an approved firm isn't in charge of the security. They don't want to have to pay someone for damage to their property or person if they can avoid it.
That's just an example where security is provided with no government. If you are asking the obvious question, that what is prevent these firms from becoming the local governments you need more detailed reading. You may also realize that providing security in Canada as an example in a lot easier than providing it in certain areas in the USA. What means is that a group of people could be less inclined to violence as compared to another group. Similarly if one is living amongst those who are libertarians, their motto of Non Aggression Principle (NAP) would imply there's less threat to violence and lesser need of protection.
Yes, you'll ask what if an external force joins hands and attempts to conquer those who don't like violence. I would suggest you may like to read Chaos Theory by Murphy if you are seriously interested. I have a feeling you really haven't read enough about the libertarian part even though you quote Plato, who philosophically was more of a socialist and a central control kind of guy.
Have a good day.
Gringo