Note: this is hasty and largely a response to the anarchy vs minarchy posts that populate the discussions here on liberty.me
In libertarian circles there are two main camps: anarchist, minarchist. Minarchists believe in small governments with limited powers. Anarchists obviously feel that government is unnecessary and unwanted.
The anarchist position can be simplified quite a bit for this particular debate, as there's about 10,000 years of government to look at. Every state throughout history has failed. Every state that exists now is merely a future failed state. No government has ever fulfilled the promises it gave to its people. Governments are responsible for more pollution, murder, and theft than any other institution on Earth. Governments generally like to devalue currencies and create boom/bust cycles. Minarchists fail to see this.
They also fail to realize that the United States of America was founded with the smallest, most limited government ever seen. Since then, many other governments have been founded with similar organizational structures. A new rule has been seen. When a small government is brought into existence with limited power, the freedom people enjoy leads to vast amount of wealth creation. This leads to large tax revenues. The tax revenues allow the government to balloon in size. As people are still somewhat wealthy and comfortable, there is no immediate push back. The government continues to swell and grab power until the people wake up, one day, in a mostly fascist state (this isn't meant to be inflammatory, please read the fascist manifesto).
This is one of those moments where I get to turn around a common statist saying. The American right loves to say "that's great on paper, but it never works in practice." Well, how's that Constitution working for you? Seems like it was great on paper and failed in practice, huh?
As a matter of fact, Constitutionalism has failed in every nation it was tried (as it's an extension of the Forum principle of law (ancient Rome, all laws were posted in the forum during the Republican period)), that's no surprise right? Republics turn into fascist states. The very first republic (Rome) is a great example. Democracies rise to great hights quite quickly and then they fall quite quickly, in a similar fashion and largely to the same result. The very first democracy is a prime example (Athenian Democracy).
Anarchists are just saying: make no design on paper. Allow organization to form in a decentralized and natural way. Governments always fail, and governments are always the worst criminals (by design and intent actually, as governments are necessarily exempt from law). Let's just stop repeating the pattern.
If you want the skinny on how things could work without a state, you need only to look to the past as well. Brehon law and other ancient polycentric legal systems can give us ideas about law and dispute resolution. Healthcare would be best served by the model of ancient China, you pay when you are healthy and when you are sick that's on the healthcare provider (Kaiser does this to some extent). Defense would be best served by the model of Carthage in which prominent traders hired mercenaries to defend Carthage, of course we could change out traders for anyone in area who's concerned about the defense of that area. Roads? Do you really think Walmart is going to allow you to not have a way to get to their stores?