Note that I'm talking about private property not personal property. Variations of the latter exist and have existed in almost every society but the concept of the former has increasingly been maximized to the point that patches of land, ways to do things, ideas in your head and sincere promises are now things you can own and exchange and people refusing to suffer hunger or lack of shelter are seen as unreasonable if they want to use what isn't eaten or isn't lived in.
Abolishing the state as long as other organizations (or individuals) are allowed to hoard resources/wealth/power seems as catastrophic a suggestion as abolishing the police as long as other gangs exist. You're just substituting one evil for another without affecting any material change.