In America "self defense even from the state" is legal but only in very limited circumstances. You can't defend yourself from a legitimate arrest for example. Self defense is not the "initiation" of force by definition right? So even then the state has a monopoly on the initiation of force.
The alternative to a state with a monopoly on the initiation of force would be more than one entity being able to initiate force, I believe that will lead to even more abuse and violence than we see currently.

I think free markets are often misunderstood as being markets without any government regulation or oversight but lack of regulation and oversight leads to market failures as well. It's hard to have a functioning free market if some of the people in the market shoot their competition or if there is no simple agreed upon third part to resolve disputes and enforce contracts. Government provides things like standardized weights and measures and civil courts that facilitate trade. Anarchists I have met would argue that private arbitration could replace courts but those are only popular now because in some cases both parties would rather avoid court, without the threat of court no one would participate in them. In the absence of antitrust laws how do you prevent monopolies from forming?