I worry about this too. I have two rental units. Tenants in one unit do have dogs, and I wonder if I should ever rent to dog owners again. I have a $3M umbrella policy in addition to the regular homeowner & fire insurance policies.
Lately I’ve been reading up on real estate LLCs. In theory an LLC looks like an ideal liability limitation vehicle, it’s just really unfortunate that the California limited liability tax is $800/yr and applies to all corporations and LLCs.