Anarcho-capitalists sometimes claim that under anarcho-capitalism, the problem of authority is eliminated, in that there are no authorities. Joseph Fetz just sounded a variation on that theme in the comments, when he claimed that under ancap, all relations will be contractual. Let us consider.
I "own" ten acres in the Poconos. It is steep land, falling past the house to a creek in the middle, and then rising to a ridge line. I probably have not been on my "back 5" in two or three years.
Tomorrow morning, I wake up to find that all government in the United States has been dissolved, and ancapistan has been declared. Tomorrow afternoon, a group of Lenape Indians arrives, and begins to erect cabins on the other side of the creek.
This prompts me to head over there. "Whoa, guys, what are you up to?" I ask them.
"Well, kemosabe, now that the evil United States government is out of the way, we are reclaiming our ancestral land. Oh, and we don't mind if you keep your house over on the other side of the creek. But we will expect thousand dollars a month in rent, thank you."
"But, but... I own this land!"
"Says who? The state of Pennsylvania? Well, the state of Pennsylvania doesn't exist anymore."
Now, it is possible that the two parties here could reach an agreement by bargaining. But if each party sincerely believes its claim to ownership, each party might just dig its heels in as well. This is why it is simply not possible for all relations to be contractual: we need some prior agreement as to who owns what before we can start contracting exchanges. (Why anyone would want all human relations to be contractual in the first place is a whole other question.)
In the case where the two parties cannot negotiate an agreement themselves, either:
1) The two parties will be in a state of war, even if no violence breaks out. ("Cold wars" are possible!)
2) The two parties submit to some authority to resolve their dispute, e.g., "The Council of Libertarian Elders," or "The Federation of Ancap Defense Agencies." And those authorities will be considering a political matter: "What should we do when two parties each have some basis for claiming a piece of property as their own, one ancient and customary, and another modern and well recorded?" And the decision they reach will be a political decision.
And it is not just land disputes that will require some authority to decide them: we can imagine on the day ancapistan is declared, that a whole bunch of people say, "Thank God! Now I can finally start copying and sharing music and movies without fear." But that afternoon, the Microsoft and Disney defense agencies ring their doorbells (with SWAT teams), seize their computers, and demand huge reparations.
Either this dispute will be resolved by pure force, and Microsoft and Disney will win because they can muster more force than can the sharers, or there will be an authority that will mediate this dispute.
People often disagree. That is a fact that cannot be wished away. If the disagreement is profound and serious, there are two ways possible to resolve it:
1) force; or
2) acknowledgment of an agent with the authority to resolve the dispute.
So, ancapistan will either:
1) be more violent than our current world; or
2) have something functionally equivalent to a government, even if it is not called that.