The incident happened Tuesday, and the ride remains closed two days later.
The man was not seriously injured and was guided to safety by Disneyland employees, according to police and Disneyland officials.
He was able to walk out on his own power, but was taken to a hospital as a precautionary measure.
Anaheim police were the first to respond to the incident, which was initially reported as someone fell out of the ride.
Police say that's not the case.
“He didn’t fall out,” Anaheim police Sgt. Daron Wyatt said. “He had to maneuver himself around the safety mechanism.”
Police say the man, who is in his 20s, has cognitive disabilities and climbed out of the coaster in the dark during a slower portion of the ride.
Disneyland officials contacted California's Division of Occupational Safety and Health (DOSH) about the incident.
Space Mountain remains closed as DOSH investigates, according to the Orange County Register.
Disneyland park officials say incidents where people climb off of a moving ride are "extremely rare."
Space Mountain is an indoor coaster that takes riders through many twists and turns in the dark.