The fire was reported at 6:20 a.m. at 925 Felipe Ave., near where U.S. Highway 101 meets Interstate Highway 280.
As of 7:45 a.m., crews were still at the scene and the fire was not yet considered under control, San Jose fire Capt. Cleo Doss said.
"We have a lot of overhaul to do, basically a lot of cleaning out of material to make sure we have the fire completely out," Doss said.
He said the units that burned are structurally unstable, so firefighters are proceeding carefully.
"Crews are having to cut through walls and things like that," Doss said. "We don't want to put any crews inside there."
Doss said all four units were full of stored belongings, but he did not yet have a dollar estimate on the loss and said the cause of the blaze remains under investigation.
Smoke from the fire was visible from the nearby highways.