Police got a report at 2.09pm local time (9.09am AEDT) about the crash in the Orange County city of Fullerton, said Kristy Wells, a Fullerton police spokesperson.
Police and fire authorities arrived on scene and battled the blaze and evacuated surrounding businesses, Wells said.
It was not immediately known what type of plane it was or whether those injured were in the aircraft or on the ground, Wells said.
Video footage from KABC shows white smoke coming from the top of a large building.
The plane crashed near the Fullerton Municipal Airport, a general aviation airport in Orange County that is about 10km from Disneyland.
It has one runway and one heliport.
Metrolink, a regional train line, is nearby, and flanks a residential neighbourhood and commercial warehouse buildings.
Fullerton is a city of about 140,000 people some 40km south-east of Los Angeles.