The owner of the car (the rental agency) will have some insurance to cover those aspects, but you should discuss with your homeowner's insurance agent whether the liability provision of your policy would cover you in the event of a serious accident.
Many insurance agencies also have a low-premium "rentals only" policy to cover those who don't have cars but rent (or borrow). Might be worth it to have them issue you one, if for no other reason than the certainty of knowing who will be ultimately responsible and have it unambiguously written.
(If you do go down that path, I'd be interested to know how the premiums commpare.)