The 1967 Outer Space Treaty clearly states that all extraterrestrial real estate "belongs to all mankind" and no state can declare it a sovereign territory. This form of sovereign ownership was and is fundamental to any claim to ownership: any government must somehow grant this right to you. Currently, land ownership is secured by a system of legal regulation, either by the country's constitution or by an international treaty that establishes such a system … in this case, this is exactly what the Outer Space Treaty does.
On the other hand, the 1967 Treaty does not make private property illegal in space or on other planets. Like any reliable legal system, the 1967 Treaty laid the groundwork, and later laws in the countries that signed the treaty built on it. For example, both Luxembourg and the United States of America have passed laws that clarify ownership of "space resources", whether those resources are on moving objects (eg, asteroids, comets) or on or below the planetary surface (like any resources that can be found on Mars … Venus or another planet) or even if these resources are solar flux, which the photovoltaic panels convert into electricity.
So, currently you can land on Mars and create your settlement: you will be vested with ownership of everything you brought with you, but not the land you dumped it all on.
But since your construction robots are raking Martian soil onto your temporary home to protect it from radiation, that soil is now the "resource" you mine and use. Now it is also your property.
Your Sabatier reactors (not nuclear, don't worry) and your plant are starting to take advantage of the thin Martian atmosphere and release oxygen, methane and water from it. You drill a well to reach a geothermal aquifer deep below your settlement and use that well to generate electricity, heat your settlement, do scientific experiments (look for "microbial life"!) And filter thoroughly to provide yourself with water supply: all these "resources "Now belong to you.
But now the situation has become more complicated. You drilled a well and obtained the rights to use it … does that give you “water rights” to the giant aquifer you used? To a certain degree? You have built so many structures in a clearly delineated area: even if you cannot own it as your “real estate,” can you not claim it by arranging everything and setting up production?
You have a launch pad nearby (but not too close) with radar telemetry around it: you have no rights to open space above the launch pad, you only own the launch pad, but you can assert those rights: your future neighbor cannot build a bridge right above your launch pad because it will prevent you from using the landscaped spatial resource that belongs to you.
Your brute neighbor might be an idiot and install an unreliable inflatable canopy next to your launch pad, since you can't point to the line that limits your property and say “don't go outside, boy,” and since that doesn't physically prevent you from using your property. You have the right to use your previously installed equipment and fry his dome. So it's not about property rights, it's about how a sane mind can deal with idiocy.
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As you can see, as soon as people start using space resources, property laws will be created quite quickly. They have yet to be created … but the underlying legal regime is clear: Mars "belongs" to everyone and therefore, essentially, to nobody.
Svetlana Bodrik