Is the sea public property?

No, ever since man started sailing and doing cartography, there are invisible property lines that belong to their adjoining countries. It is a part of a country’s defense.

Territorial waters, or a territorial sea, as defined by the 1982 United Nations Convention on the Law of the Sea, is a belt of coastal waters extending at most twelve nautical miles from the baseline (usually the mean low-water mark) of a coastal state. The territorial sea is regarded as the sovereign territory of the state, although foreign ships (both military and civilian) are allowed innocent passage through it; this sovereignty also extends to the airspace over and seabed below.

Oceans, seas, and waters outside of national jurisdiction are referred to as international waters. Also called the high seas.

Ships sailing the high seas are generally under the jurisdiction of the flag they carry and display.

So I guess you could build a permanent colony in the High sea’s and set up shop