I had the possibly unprofitable hypothetical idea of starting a mobile recycling business where a truck comes to you and pays you door-to-door for your recycling at below market prices. Can this be done legally in California?
At minumum, you're going to need local (and possibly state) businesses licenses, a business structure (like an LLC) that protects you and your employees from liability, and business insurance for you, employees and vehicles.
Specific to the recycling business, you may possibly a license or permit to buy and haul recyclables and to either process them in your own warehouse or storage area, and that may not be something you can get, depending on local and state laws dealing with trash and recycling companies.
Start with your local county/city business permit office to see what you need to do. Google for recycling companies in California and see what you can find out.
Can you buy second-hand goods in order to resell them at a profit in California? Yes.