Police did not identify the suspect or say if he was a UPS employee. They told a news conference the incident was not terrorism-related and they recovered two firearms from the scene.
Two other people were shot and have been taken to an area hospital, police said.
The shooter and the victims were all drivers, said Steve Gaut, head of investor relations at UPS. The incident took place while the workers were gathered for their daily morning meeting before going out on their routes, he added.
Gaut said the facility’s employees have been released from work and he believes most have left the building. The company is providing trauma and grief counseling to employees.
Live video showed a massive police presence near the facility, with workers being led out and embracing each other on the sidewalk outside.
"The company is saddened and deeply concerned about affected employees, family members and the community we share. Our thoughts and prayers are with all those touched by this incident," UPS said in a statement.
Victims were taken to the Priscilla Chan and Mark Zuckerberg San Francisco General Hospital, spokesman Brent Andrew said. He said he could not say how many patients were taken to the hospital or give their conditions.
In 2014, a man shot and killed two of his supervisors before turning the gun on himself at a UPS distribution center in Birmingham, Alabama. The gunman had recently been fired from the facility.