A couple of things you must understand before talking about “AnCap”:

The state, in this day and age, is run by the capitalist class.

“The state is, therefore, by no means a power forced on society from without; just as little is it ’the reality of the ethical idea’, ’the image and reality of reason’, as Hegel maintains. Rather, it is a product of society at a certain stage of development; it is the admission that this society has become entangled in an insoluble contradiction with itself, that it has split into irreconcilable antagonisms which it is powerless to dispel. But in order that these antagonisms, these classes with conflicting economic interests, might not consume themselves and society in fruitless struggle, it became necessary to have a power, seemingly standing above society, that would alleviate the conflict and keep it within the bounds of ’order’; and this power, arisen out of society but placing itself above it, and alienating itself more and more from it, is the state.” - The origin of the family, private property and the state by Engels

Now to answer your questions:

Can you explain exactly why do you think that AnCap society would not have ownership ?

I didn’t say that. How would it be capitalism if the capitalist themselves didn’t exist? The characteristic of the capitalist class is that it has private ownership of production, it pockets the products of the processes of production. How is this ownership enforced? With violence, laws are made to protect these relations of production and when broken violence is used to enforce them. Who enforces this ownership? The state, through police and military.

Also can you compare that to the current system in real life?

“AnCaps” want the same society we have but without the state, in their twisted mind the state is some commie plot and must be gotten rid of.

What would happen if the state magically disappears? There would be no one to enforce private property, the capitalists would have to enforce these relations themselves by turning into some sort of warlords, imagine a world where each capitalist needs a private army to enforce their property (who would stop their army itself of taking over?), it is literally going back to the past it is comically dumb.

Another point is that AnCaps are grifters, see Milei in argentina for example. He claimed to be an AnCap but he is doing everything in its power to turn argentina into a police state run by the US!

From: https://www.nakedcapitalism.com/2024/04/musical-chairs-in-latin-america-argentina-applies-to-become-global-partner-of-nato-while-colombia-expresses-interest-in-joining-brics.html

In those two months, the Milei government has signed a memorandum of understanding with the United States allowing members of the US Army Corps of Engineers to operate along the Paraná-Paraguay Waterway, including large parts of the river Plata basin, upon which roughly 80% of all Argentine exports travel. It has agreed to buy $300 million of second-hand, 40-year-old F-16 fighters from Denmark, with the help of US financing. It has also announced the establishment of a US naval base in Ushuaia, on the southern tip of Tierra de Fuego, often described as the last stop before Antarctica.

Accused of selling out his country’s sovereignty for seemingly nothing in return, Milei argued, with a straight face, that allowing the installation of a US military base in Usuahia is the greatest act of sovereignty of the past 40 years since it will strengthen Argentina’s territorial claims over the Antarctic.