Pet care work carries real liability exposure, an escaped pet, a dog bite, property damage during a home visit, or an injury to the animal in your care can all lead to a claim. Two coverage types come up most often for pet sitters and dog walkers.
General liability / commercial pet care insurance covers third-party injury or property damage claims, this is the baseline most serious pet sitters carry, and many boarding facilities or apartment complexes require proof of it before granting access.
Bonding protects against theft or dishonesty claims (e.g. if something goes missing from a client's home while you have access), it's a different kind of protection than liability insurance and is often sold as a package with it by pet-sitting-specific insurers.
Whatever coverage you carry, disclosing it clearly to clients, and getting emergency vet contact info and care instructions in writing via a pet sitting agreement before the first visit, are the two things that most reduce both your actual risk and your liability exposure.