I must prove, beyond a shadow of a doubt, that only the state can provide these services, eh? It is actually easy for me to see it, so I'm not sure at what level you are, but I'll try. Just like a shoe store provides shoes, the state provides law, and government, which protects property.
The very first concept to understand about the necessity for the state is property. Individuality generally and property specifically. Land rights are at the heart of the origin of the state. Kings ruled over land and the resources contained in his land. Land owners are kings of their own castle. Just like kings protected their land ... homeowners must also protect their land.
The State is a monopoly of force in a given territory. The minimalist state is individualism. As individuals we have the duty to protect our own life. That is ours and ours alone. (i.e. we have a monopoly of force in our given territory.) In other words, however far we can swing our arms to protect ourself. If an individual has a gun then that territory expands to the limits of the gun. The individual with a gun expands his monopoly of force from the swing of his arms to perhaps a few hundred yards. His/her monopoly of force has expanded to a greater circumference. If a group of individuals get together and build a nuclear weapon, then the monopoly of force expands even greater perhaps for miles upon miles. If that nuclear weapon is put on a rocket then the monopoly of force expands even greater still and can become offensive rather than defensive for hundreds or thousands of miles. Indeed, any individual can become offensive rather than defensive. An offensive individual, or group of individuals, must be restrained in order to protect liberty. So, the weapon can be used, by individuals, or a group of individuals, for defense or offense. If used for defense, then it, protects liberty, and is a legitimate state. If it is used for offense, then it, fails to protect liberty, and is an illegitimate state.
When land is claimed to be "owned" by someone, then along with the ownership claims comes rights. For example, the right to exclude anyone from trespassing. (i.e. no one is allowed to come into your home without your permission... even... no one is allowed to set foot on your property without your permission.) Who protects that right? The land owner. Who backs the land owner up if he/she kills a mean bully who steps on your land and rapes you? The State. That is the purpose of the legitimate state.
Mises said it best,
We call the social apparatus of compulsion and coercion that induces people to abide by the rules of life in society, the state; the rules according to which the state proceeds, law; and the organs charged with the responsibility of administering the apparatus of compulsion, government.
In other words, the state says, by law, NO mean bullies have the right to rape another human being. If a mean bully comes into your home and tries to rape you of your person or your property, then the state will back you up when you stop him. You have a monopoly of force in your given territory.
That is the purpose of the legitimate state. The protection of property, liberty, and peace.