What you are looking for is called "non-owner car insurance." It is for people in exactly your situation: someone who does not own a car, but frequently rents or borrows cars and needs liability coverage.
Not every insurance company offers it, and it is not available in every state. In addition, if you live in the same house as someone who does own a car, you are probably not eligible to purchase non-owner insurance. So you'll have to do some more investigating to see if you'll be able to get this. Here are a couple of articles about the subject:
- NerdWallet: Non-Owner Car Insurance
- Barkrate: No car? You may need nonowner car insurance