Most construction and household debris is fine. The limits come from California's rules on household hazardous waste and from what the City and the county authority will handle.
Hazardous material always stays out: paint, solvents, fuel, used motor oil, pesticides, pool chemicals, fluorescent tubes, batteries and anything with asbestos in it. California directs those to household hazardous waste collection rather than to a container, and the authority runs collection days for them.
Hanford's own excluded list is wider than most. The City will not collect ammunition or combustible material, dead animals, hot ashes, infectious or medical waste, oils, axles, heavy machinery parts or large appliances, and it will not collect rubble through its regular service.
There is one rule that makes care worth the trouble. If prohibited material reaches a City collection vehicle, the City treats the entire truck load as contaminated and charges the person who put it there for disposing of the whole load. The same thinking applies at the authority's entrance, so one wrong item can be expensive.
Unsure about something in a shed or a barn? Check with your account manager before it goes in the container.