The easy answer is no. You are not required to carry insurance to own a small business. It is recommended, but not required. (There are some exceptions for businesses that must carry insurance, but if you are in one of those businesses, you know it.)
Of course, there are no hard and fast rules. If your business owns a car, you need to have auto insurance. If your employees are going to drive for you, you SHOULD have auto insurance. If you drive your own car for business, you SHOULD have auto insurance.
Why? The first one is easiest. If the business owns a vehicle, you must insure it because California law requires the owner of a car to insure the car. So, you must insure it. If your employees are going to drive for you, you do not need insurance, but if they drive their car, and they are not adequately insured, the person they hit may sue you. If you drive your own car, and it is insured, you should have auto insurance because your personal policy may not cover you for damage that occurs when you are in an accident while working.
Insurance is one area where you may want to hire someone to help you figure out what insurance is required, what you should have, and what you do not need.