Can anyone start a pearl farm?

There are different rules in different countries so my comments on this subject pertain only to the Cook Islands. The short answer is no. Foreigners cannot own land in the Cook Islands and this also pertains to parts of a lagoon where you would establish a pearl farm. However, the rules go further. You can only own land (part of a lagoon), if you have genealogical ties directly to that piece of land (lagoon). For instance, although they are both Cook Islanders, someone of Rarotonga (a southern group island) descent cannot own land (lagoon) in Manihiki (a northern group island) and vice-versa.