You can obtain liability insurance if you do not own a car. It is often called "non-owners coverage" and will generally provide protection regardless of the car you are driving.
However, you must have an insurable interest in the vehicle in order to get physical damage coverage. An insurable interest is a "stake" in the continued existence of the vehicle. Therefore, if you are a co-owner of the vehicle, loaned money on it, or otherwise have a financial interest in it, you have an insurable interest sufficient to support an insurance transaction.
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