It is important, I think, to understand what one should or should not be an anarchist against. Being in anarchy against what is right in "the state" is just as wrong as being for anarchy when it is wrongly applied and misdirected.
For instance, we recently lost our home to property taxes we could not afford. I can tell you, after half a year of forced homelessness as a result (at the age of 65), along with my elderly wife (61) and 24 year old handicapped son, homes or private property are essential to life, peace, happiness and the pursuit thereof.
Yet neighbors think they have the "right" to vote other people's property as collateral for bonds, to be repaid yearly in the form of "property taxes," thus allowing the courts to rob and wrest homes from the economically disadvantaged made poor by other police-state actions, such as asset forfeiture (theft) without any evidence, proof or existence of any criminal wrong-doing whatever (other than by robbers we now call "the cops").
Meanwhile, ownership of property and tax exemption by legal fictions called corporations, which in turn use the lucrative economic base this affords them to avoid or barely pay minimal property or income taxes, while influencing politics in their favor, and sending profits out of local areas to corporate headquarters (which may be in other countries), thereby robbing the community/country of the velocity effect of money changing hands a sufficient number of times and remaining within a local economy long enough to sustain its buying power; these things should be seen as grounds for true anarchy against very real wrongs and evils, such as so-called "eminent domain" property seizures on behalf of corporate interests without adequately compensating the owners.
Ownership of property itself is not the problem, since you possess even the various parts of your physical body, and have every right to defend them against attack, loss or harm. In like manner, property being merely an extension of the use of those parts to create order out of chaos and greater value out of raw materials obtained by the sustained exercise and hard work of those parts, deserves equal protection from thieving neighbors as well as greedy politicians, common criminals, and actual (not feigned or faux) foreign or domestic enemies, at home or abroad.
The house you live in is personal property. Nobody should be able to throw you out of it.
it's not though....assuming you take a mortgage out to purchase said property, the person who gave you the money is the owner until the money is repaid...