Do I need a physical address for a Google Business Profile?

No. If you travel to your customers, Google lets you register as a service-area business, and you hide your address while still showing the towns you cover. Most plumbers, roofers, HVAC companies and landscapers should be set up exactly that way. You still have to give Google a real address during verification, but it does not get published.

Two things trip contractors up here. Using a home address and leaving it visible is the first, since it publishes where you live and does not help you rank. Renting a mailbox or a virtual office to fake a location in a town you want to rank in is the second, and it is a fast way to get the profile suspended. Set the service area to the places you genuinely drive to. More on getting the profile right in how to optimize a Google Business Profile.