Established in 1965, the school is named after rocket pioneer Robert Hutchings Goddard.

Walker Air Force Base, located on the south side of Roswell, was considered a major target for Soviet ballistic missiles.

The classrooms were built underground so that the school could serve as a bomb shelter in the event of nuclear war.

The gymnasiums, cafeteria, auditorium, and other larger rooms were built above ground, on top of the classrooms.

GHS competes in the New Mexico Activities Association, as a AAAA school in District 4.