I was planning to do the move myself, with a rental truck. I'm not confident that rental trucks will be available. I'm not sure I'd want to be in one anyway -- even before the pandemic the sanitary condition of rental trucks was dubious at best.
I'm hoping that moving will be considered essential travel, although tonight is just before the start of the new policy. If a lot of people ignore the "shelter in place" during the next week, the enforcement will probably get stricter.
I don't have an answer for you but it's a real problem. Paying double rent (or rent plus a mortgage) involves a huge amount of money in the Bay Area.
I'm probably going to try to contact the county government tomorrow and get an answer but I'm sure I'll be the millionth person on hold on the phone.
(Also, guess who needs a smog check to register his car and might not be able to get one?)
You should be okay on the smog check, because auto repair shops are considered essential in the order.