So, I’m not an anarchist any more, but just to throw in a few odds and ends:
In the socialist conception of things, the state is the network of social forces that separates classes. Things like cops, or parliament are big obvious parts of “the state”, but things like CNN or Microsoft are too, despite being in private hands. In your hypothetical, the apparatus you use to accumulate wealth is “the state”.
A part of this reading is that it requires active effort to maintain the state. To take an example, let’s say your method of wealth accumulation was by becoming a landlord. You own the land and dwellings that people shelter in. Cool.
What makes this “yours” and not the tenants? Well, you paid for it, yes, but unless you spend all your time debating your tenants about the philosophy of ownership, you’re going to need enforcers. Enforcers who take your claims of ownership seriously. And this gets more and more necessary as you get more land and more tenants. You’re not going to fight ten tenants yourself to extract rent, let alone 10,000. After all, one would expect a slave to try to escape even if you rightfully paid for them, why not housing or food or anything else people need to live.
In most strains of anarchism, hoarding property so you can exploit your fellow man is violence, just as in liberalism walking across an empty bit of lawn that someone owns is violent.
But then, this isn’t really my beliefs any more, so um… idk, I hope it helps