Terraspheres were a series of simulators built by Species 8472 in 2374, designed to mimick conditions of Starfleet Headquarters on Earth. Overall, there were approximately a dozen terraspheres scattered throughout the Delta Quadrant.
Terraspheres were heavily fortified, and were protected by an unknown type of forcefield and shields. In all, they were powered by 13 thermionic generators located beneath the surface of the simulated city.
The polymorphed members of Species 8472 were conducting a role-playing mission to better understand their enemy, the Federation and Starfleet. Similar in principle to Soviet Union attempts to recreate conditions in the United States of America, for the same purpose, during the Cold War.