As to being in an 'anarchist society,' I'm uncertain as to what you mean, exactly.
I can tell you this, as anarchist who desires a deepening sense and expression of anarchy, I have no use for your tools which aid in creating and maintaining such a worldview as believing lawns and lawn-tools, are in the least bit necessary or desirable.
Since you wish to begin your thought-experiment entailing a budding Ancapistan within an actual anarchist milieu, I'd probably answer that If you try to fence me in by encircling me with your 'private properties', oh capitalist prick-tater, I do believe I'd fight you, rip-down your fences, and if I couldn't do it myself, I'd become affiliated with others to kick your ass away from me and all I love, given the poisons capitalism has offered in so many forms: slavery, enclosure, ecocide, genocide, colonialism; all grounded in a pathetic worldview where all relations become reified "things' to be bought, sold, destroyed, as long as it all adds up nicely to those who steal the most.
Which brings me to:
It's not so much that anarchists 'cannot tolerate' capitalist (anti-)social relations, as much as capitalist relations are neither social, nor do they even come close to hinting of 'anarchy.'