After serving in the Marine Corps, Musgrave earned an impressive list of academic credentials, including bachelor's or master's degrees in mathematics, operations analysis, chemistry, literature, and physiology, as well as a medical degree from Columbia University (1964). In 1967, as an expert on cardiovascular and exercise physiology, he was picked by the National Aeronautics and Space Administration (NASA) as one of a group of scientist-astronauts to serve on future space missions. While training as a jet pilot, he began designing space suits, life-support systems, and other equipment used for extravehicular activity (i.e., space walks) on NASA missions, a field in which he eventually became preeminent.

Musgrave's first space mission was on STS-6, the maiden flight of the Challenger space shuttle, in April 1983. He then served as flight engineer on the crew of Spacelab-2 in July 1985, a mission that performed astronomical research. He was a mission specialist on STS-33 (November 1989), STS-44 (November 1991), and STS-80 (November–December 1996). His most important mission came in December 1993, when, as payload commander on STS-61, he led the crew in a successful effort to repair the faultily constructed Hubble Space Telescope. Musgrave also served as capsule communicator (i.e., the ground-based communicator with crews in space) for many Skylab and space shuttle missions, and he published many scientific papers on aerospace medicine, exercise physiology, and other subjects.