To some degree, all property rights are socially constructed; some property rights are, however, more socially constructed than others. Being socially constructed is what makes some state of affairs a matter of rights as opposed to the pure individual exercise of physical force. (Alternatively, one could label individual force primitive or irreducible property rights, in which case we can talk about the alternative of complex, social property rights. The distinction remains, regardless of the label.)
We speak of a right when a group of cooperating individuals agree to organize their power of coercion to a specific end. An individual has a right (or a social right, as opposed to a primitive right) to the property she physically possesses, for example, not just because she can physically control that property, but also because any individual attempt to take the property will be met by the organized resistance of the community. This organization has to come about by some sort of mechanism establishing an agreement between multiple individuals; in this sense all rights are therefore socially constructed.
If we label the organization of coercion as "government", then by definition all rights come about because of government infringement of individual freedom. We have a specific right to life only because the government infringes upon people's individual freedom to kill each other. We have a right to possess property only because the government infringes on the individual freedom to take stuff as one pleases.
Furthermore, while it is logically possible, there are in practice no rights at all that are established by universal explicit agreement, without the coercion of some dissidents. (Well, perhaps they are, but I can't think of any.) The right to life is important because there are people who, without the threat of coercion, would in fact kill others; there are those who would otherwise take others' things.
Thus the notion of the free market is even at this basic level necessarily an approximation, an idiomatic usage of "free". For there to be any sort of market — in the ordinary sense of the word — there has to be some limitation of individual freedom enforced by organized coercion. Without such government interference, there wouldn't be a market, there would be only a battle.
The notion of the free market gets even more idiomatic when we consider physical versus abstract ownership.
Physical ownership is established by physical possession, control, use and/or occupation. I occupy and use my house; I control and use my car; I possess my books even when I'm not using them. Social construction is not necessary to establish physical ownership. Even an alien observer from Sirius, completely ignorant of our language or symbolic communication, could determine physical possession by observation. Even though rights to physical ownership are socially constructed, the fact of physical ownership is, well, physical.
Abstract ownership, alternatively, is ownership that is established by social construction. The fact that my landlord "owns" my house is completely socially constructed; there is no physical fact at all that establishes her ownership, even though I physically occupy and use the house. (The deed records ownership, it doesn't physically establish ownership.)
Related to the notion of abstract ownership is abstract property, property with its very existence due entirely to social construction. The most obvious and pervasive example of abstract property is, of course, money. The value of my bank account or the dollar bills in my pocket, or even the gold in my safe-deposit box is not a matter of the physical value of what I own; it is valuable only because most everyone agrees that it has value. Naturally, all abstract property is abstractly owned.
Just as rights to physical property have to be enforced, abstract ownership has to be enforced as well. My landlord owns my property only to the extent that the members of society (notably the armed sherriffs) will physically evict me from the premises if I don't pay my rent.
Thus any market in modern society requires not just government interference to establish rights, but it also requires government interference to establish abstract ownership and to create abstract property.
I'm not objecting to the concept of abstract ownership per se. Money is a terrific invention, I don't object in principle to paying rent, and there are no end of uses of abstract ownership that have obvious pragmatic value. It is a contradiction, however, to treat abstract ownership as if it were physical ownership, specifically to argue that because some right pertains to physical ownership, it must therefore necessarily apply to abstract ownership.
TANSTAAFM: There Ain't No Such Thing As A Free Market.
To have a market in the first place, there has to be some form of government infringement on individual freedom. As the old joke goes, "We've established what you are; now we're just arguing over the price." The choice is not whether to have a "free" market, but rather how much and in what ways to restrict freedom to have a market, And more importantly, who decides how and how much to restrict freedom. All the varying theories of political economics are different positions over who gets to decide how the market is to be restricted.
Capitalism is the idea that the owners of capital should decide how to restrict the market. But capital itself is an instance of of abstract property; both the property itself and its ownership is socially constructed, the result of nothing but agreement. Thus to remain internally coherent, Capitalism must necessarily enforce the value and ownership of capital.
Once the the value and ownership of capital is enforced, and the decisions about what should be enforced is handed to those who enforce capital's value and ownership, they will, as we expect of all self-interested individuals, do whatever it takes to retain their position as those who decide what's enforced. The individual freedom to change the system and assign privilege to a different group will always be coercively restricted. Privilege, once granted, is self-perpetuating.
Human beings typically have a narrow view of who qualifies as equals, as "real" human beings. More importantly capital (like many other resources, physical and abstract) is more effective when it is concentrated rather than diffused. Therefore, a capitalist social system will inevitably concentrate capital and the privilege its ownership entails. A capitalist society is inevitably elitist. Not the irrelevant elitism of individual merit, but rather the elitism of self-perpetuating privilege concentrated in a narrow group, concentrated by virtue of having had privilege concentrated in a narrow group.
It may well be the case that capitalism is the best we can do. There are some features of capitalism, notably that capital has to actually be used to be effective, and what can be used risks being lost. There is at least a degree of meritocracy within the capitalist elite. Furthermore there are avenues of entry for the arbitrary individual into the capitalist elite: One need only accumulate capital and use it effectively.
The above considerations are pragmatic and at a high level of abstraction. The notion that Capitalism is intrinsically justified, or justified by our notions about freedom or about physical ownership, are at best fantasies and at worst outright lies.