Well, not everybody calling themselves anarchists understand the economic realities of it.
Of course you can keep your stuff, the fruits of your labors(though i think you might want to trade them for other stuff), and you can be as 'rich' as you want.
Me, personally, i want a corvette and a neon sign shoppe.
Other than that, supper and a bed is good enough.
In an anarchic world stuff will be valued more appropriately with family time, leisure, and just plain living that we miss out on today in our two earner world.
Here is one fella's view of it.
Here is another.
I would venture that most people who call themselves anarchists are not very well versed in economics at all.
Me, personally, I want a fairly large area of land where nobody else is allowed without my explicit permission and invitation.
I think most of us want little more than that.
Maybe the kids, but certainly the mature just want to be free to do as they please.
There isn't a single country without private property where people can do as they please. In fact, the lack of private property tends to go along with being forced to stay put and provide your labour at gunpoint.
Rule by force is the disease, who and how are symptoms.
I think in the absence of being able to point guns at folks and tell them what to do, crapitalism goes away.
What I offer is a much better deal.
It isn't a new idea.
Here is a version from 1887.