there are many different visions of anarchist society, but i don't know of any of them that envision people having to continually (or even usually) fight to keep their stuff safe.
some of the issues that are brought up by this question (as i see them, and many anarchists have discussed them)...
a. personal property is different from other kinds of property (ie, your toothbrush or treasured photo of your sister) is different from something like a house, or even a hammer, which you only use periodically, or which other people can use with you.
b. different groups of people would figure out how they want to live with each other. so you could find people who agree with you about how to value things, and how to protect those things, and live with them.
c. in a different world (one where we hadn't been taught all our lives to value work, or laws, or to define ourselves through what we own, for example), we would care about things differently, so that the things that are important to us now, would be less important, and things we don't even recognize now (or don't find as important) would gain in value.
an exercise in imagining one different world is a book that gets referenced here all the time, called bolo'bolo (by p.m. and published by autonomedia, originally). it might be a fun read for you.