The anarchist position is that the goal is to have no government, only private businesses providing all services currently handled by governments, including and especially police, courts, and military defense.
Each person would voluntarily pay for whatever amount of each service they think they need, contracted with whatever company or combination of companies they choose. Over time, these companies would develop agreements on how to work with each other when needed and would rise and fall individually depending on how well they did at actually providing those services.
Criminals will pay just restitution for their crimes as decided by independent arbitration, and any other penalty that the arbiter thinks is just and will be carried out and enforced by the security company you or any arbiter contracts with.
Civil complaints will also be decided by arbitration with the same understandings.
All of this will be handled via universally and voluntarily accepted just means without infringing on anyone else's rights and no force will be needed or used against anyone but the offending party in criminal cases or losing party in arbitration cases, and no security company will interfere with enforcing the judgments reached by an arbiter or by another security company, who are all held to the highest standards by their contractees.
People would be motivated to interact peacefully because their overriding concern is about maintaining their reputations so that others will be willing to keep interacting with them. If there's a danger, people will voluntarily band together to do whatever it takes to defend what is right and just.
More just? Yes.
Possible? Only if you can successfully get from here to there where you have multiple competing institutions, each of which has to be individually capable of maintaining the strength needed to successfully defend everyone from enemies both internal (criminals) and external (other countries, gangs, mobs, etc.) within their purview for both the current and foreseeable future, as well as the actions of all security companies and all arbitration settlements being universally recognized as definitive and enforceable by all parties that could possibly be involved on any side in any dispute.
But human nature isn't going to change and neither is the rest of the world, even if you could get any part of it to change over.
That is what I see no possibility of, for all the desirability of the end goal, and what no anarchist has made even a stab at actually laying out, step by step, that I've ever read or heard.
I read that these institutions will develop themselves … but how does the change over actually take place? Saying that it just happens somehow and everything else is going to pause while it's going on doesn't cut it.
I read that everyone will accept the arbiter's judgments and the actions of these security companies as all being beyond reproach, that people are only going to pick companies based on what's right, not necessarily what's best for their own personal immediate interests, that there will be no conflict between these sterling characters and businesses … but that's not how real people act.
I read that everyone is going to realize what is actually in their own best long term interests and be willing to pay for it … but they haven't been up until now. What fundamental part of human nature is going to change and what will change it?
In some ways, anarchists are a mirror image of the statists. If everything just worked the way it's “supposed” to then everything would be just wonderful.
But it never does. People persist in being the same kind of people they've always been. Reality doesn't change simply because someone thinks that's the way things ought to be.
Accepting the realities about human behavior, desires, and motivations has to be the bedrock of any human endeavor that has even the least chance of success. That means accepting that the bad is just as real and just as basic as the good.