TL;DR: Basically, in the US at least, Libertarians are spoiled white guys who don't even understand how good they have it and have Ayn Rand power fantasies that they'll make their own way and the rest of the world has just been dragging them down.
A couple of my college buddies are full on Ludwig Von Mises/Murray Rothbard anarcho-capitalist nutjobs. The basic conceit is that all governments and states are illegitimate uses of force and are drags on the free functioning of the economy. Left with no "coercive" governments, people will competitively self-organize into private collectives to replace all governmental services, and all resources will flow to their best and natural use. It's absurdly naive and ignores absolutely everything about human nature and even the de facto reality of their desired end state.
So somehow private property will continue to exist and be protected by voluntary courts and security, and funny how it works out that in this case my buddies get to keep the fruits of the privilege enjoyed by centuries of their ancestors and built up in a decidedly non anarcho-capitalist system. All existing government property will be sold off and the proceeds distributed to... someone? No word on how natural monopolies like the best water route between two river ports will be handled, but it will be privately negotiated and definitely perfect!
It will be a utopia of people pulling themselves up by the bootstraps and not letting silly things like "personal safety" or "living wage" or "stewardship of resources" get in the way of making the completely even-handed and non-coercive deals that all people will make with the private entities that spring up to replace governments, but only VOLUNTARILY! People definitely won't make deals they don't like, and that reduce their future power, to avoid death in a "market" with limited opportunities. They definitely won't leave their shares (or whatever) to their children and recreate all the same social structures we have now, but with corporate self-interest as literally the only governing norm.
Now, I suppose you could end up with corporate bodies that are outcompeted by "fairer" competitors (ignoring, of course, all first mover advantages and the willingness to protect profits by violent force that we already see in so many times and places), or maybe certain security and judicial corporations will make agreements with each other and install themselves as a layer over the more economically productive companies and collect fees that are definitely not taxes. Maybe some of them will be the "fairer" entities.
But where does that leave you? Basically, our current world is already at least a little better than the libertarians' best-case scenario, and what their system really does is tell people to give up, that they are not worth one cent more than the economic value they can provide to someone else, and that they deserve no voice in the governance of their lives beyond what they can take.
How this doesn't descend into competing warlord fiefdoms, eventually to be swept away by spasms of violence (in this system, "competition" is just a euphemism for politics and war), is beyond me. With some luck, it might lead to some parts of the world on a tortuously slow and uneven march in the vague direction of egalitarian governance to moderate the use of coercive force. In that case, CONGRATULATIONS! You've landed the world right back where it started, but now with millions dead and the Earth in even worse shape than it would have been.
