The state is needed to enforce private property, not the lack of it.
Critically, you assume that authority is required for order, so obviously you're unable to envision whatever it might be that anarchists are getting at.
Sure, communism is an ideal that does not exist in practice currently, but the point of an ideal is to work towards it. The "anarcho" part specifies complete opposition to authority in praxis as well. Strategies could include unionising, community building, mutual aid, permablitzing, FOSS, copyleft, and whatever else can undermine the current power structures while maintaining anarchist principles. Which explicitly excludes Marxist-Leninist strategies of coopting the state, or forming any other kind of heirarchy.