That's where you are wrong. Free markets can provide redress for when people harm one another. Libertarianism doesn't simply fall apart once somebody does harm, it simply is the axiom which forms the structure of the system, not the key to it working.
It may be the axiom... so what then IS the key? Wishful thinking? No. A promise, or covenant as Hobbes masterfully explained, between the people will not work in the state of nature, ie, without governmental authority. Only government can successfully form a social contract and enforce it. You can not form a society based on people's promises to each other. This is so elementary and libertarians are at extreme odds with the greatest founding minds from Jefferson to Madison to nearly everyone else. A community -country- must turn on a common center.
Government cannot serve people better or more morally than a free market, government can only make you serve it. Any perceived efficiency all comes at the cost of government's coercive control and legal extortion of the citizens that it controls.
We do not give up our rights, they endure always and forever, but their reach my be shortened depending on context. We do not give up rights, we give up some power. We empower government to protect rights, after all, what good are natural rights when you can't fulfill them? Nothing is easier broken than a man's word... you need an enforcement mechanism to make our words true. You have liberty in nature (or this magical free market you speak of) but you can't enjoy it or preserve it. You have to join society to create a life worth living.
The only laws that are needed do not need government to provide them. They are self-evident, that each person is their own master and they are the master of their labor. Every just law is derives from that simple natural law.
There is no real evidence anywhere that having no government would improve people's lives but there is nothing but evidence showing government can create the conditions where people become much safer, happier, and prosperous. Governments that can properly align to and exploit the free market tend to do much better at this.
I suggest you read some libertarian works. I recommend For a New Liberty by rothbard.
http://mises.org/rothbard/foranewlb.pdf
I am familiar with Rothbard. Although I lean libertarian in many ways, I do not call myself a libertarian and I don't subscribe to much of his disturbingly foolish interpretation of American history (from his Revolutionairy revisionism to his worship of Jackson to his hatred for Lincoln). The bottom line is the libertarian fixation with liberty as an end in itself and liberty for liberty's sake is morally bankrupt. Liberty is awesome, and should generally be maximized whenever possible, but the reason for doing this is because it's more in line with reality, natural law, and creates societies that improve human life. It may seem counter-intuitive, but we need some government to maximize our functional liberty and I'm not going to believe libertarian talk that the state of nature, or free market, or whatever you want to call it, can improve people's lives better simply because, well, you say it will.