3,200 square feet at a landing strip in Aruba. Others are massive. The Ramstein Air Force Base in Germany covers over 3,000 acres with over 10,000 personnel.
Most make no military sense. We maintain over 50,000 troops in Germany even though the threat of Soviet tanks storming through the Fulda Gap is long gone. The Pentagon has 32 bases and facilities scattered around the small island of Okinawa that have no purpose except as World War II trophies.
The purpose is the same as why the Roman Empire established fortifications at the far reaches of their known world. The United States is projecting its power around the world to threaten and intimidate. And we do it out of fear. The rest of the world scares us.
See also America's Empire of Bases by Chalmers Johnson
The World Wide Network of US Military Personnel