You can get a California driver's license without a social security number (SSN). You were correct that California requires a social security number to get a California driver's license; however, the California Department of Motor Vehicles listed on its site an exception to that rule.
The California DMV notes that you are exempt from California's social security requirements if you are legally present in the United States and are ineligible for a social security number. If this is the case, to get your driver's license, California will you require you provide them an acceptable birthdate and legal presence document.
So first check with the DMV to see if you can obtain a California driver's license since you do live there on and off. If you are unable to get the driver's license, it will be more difficult to get insurance since most all car insurance providers require you to have a US driver's license to rate you on.
Without a US license you will need to check into getting specialized auto insurance there are some companies such as The Sunrise Group that offer tourists to the US coverage for their vehicle. With this company you may purchase coverage for Liability only or Liability with fire, theft, vandalism and Collision. This insurance is for private passenger automobiles, motorcycles, private passenger trucks (under 1 ton) and motor homes.
Refer to our California car insurance page for state-specific minimum requirements, average rates, laws, and other helpful tips about insurance in the Golden State.