You may be able to buy a car and insurance in your own name at age 16 however you would normally be required to have a parent or guardian sign with you on any legal documents surrounding these items. The minimum age that one can buy a car and title it in their name and then obtain car insurance will differ depending upon state laws and insurance companies' guidelines.
In general one must be an adult to obtain an insurance policy on their own, so over the age of 18. If you are under 18, and thus still considered a minor, then normally state laws and insurance companies guidelines will require you to have a parent or guardian sign on legal documents, such as an insurance policy, with you.
Typically if you are old enough to have a driver's license or own a car you can buy an auto insurance policy. If you are a minor, your parent or guardian will usually have to sign on the insurance contract with you though,
If you are a minor and what to own and insure a vehicle you should check with your state's Department of Motor Vehicles and insurance regulator to see if there is a minimum age set and what extra measures you may need to take until you become 18 and considered an adult.