If you were storing a few bags for one day, or one bag for a few days, you might have more options. To leave a few bags for a few days, however, you will need to do some legwork.
The first place I would check would be with your hotel in San Francisco. Some hotels are more willing to do this than others, and storage conditions may vary widely— a locked room in some cases, stacked in a corner behind the desk in others. It is extremely common for guests to ask that bags be held for a few hours on the day of checkin or checkout; asking for multiple days is worth a shot, especially if you are staying there for several days, and most especially if you are staying there before and after the excursion. Generous tips (before and after) to the concierge or bell desk may help.
The second place I would check is with the motorcycle rental agency. You're hardly the first people to not want to lug suitcases around on the back of a bike. I know EagleRider and MotoQuest explicitly do offer luggage storage as a complimentary service.
I would also check to see if any private clubs you are a member of have reciprocal relationships with San Francisco-area private clubs, and if they have facilities you could leave the bags at. For example, members of the VSC can request an invitation from the Marine Club. It's a thought.
But there's little incentive for a company or other organization with no relationship with you to accept. They don't want to store, or be accused of storing, anything dangerous or illegal, and they don't want responsibility for the bags or anything in them being stolen. The only public left luggage facility in San Francisco proper, so far as I can tell, is at the California Welcome Center at Pier 39 (toward the eastern end of Fisherman's Wharf), which charges $5 per bag. The lady with whom I just spoke on the phone, however, says they do not permit overnight storage.
You could call around to other hotels and hostels and see by trial and error if anyone will take them on. The Chancellor Hotel comes up in some web searches as an option. The Pacific Tradewinds Hostel explicitly offers storage to its guests. Now, maybe you don't care to stay there, but at $32/night, booking a bunk that stays empty would still be cheaper than the airport ATA.
The temporary Greyhound terminal and Emeryville Amtrak station say they offer luggage storage facilities, but I could not find any information on the services, prices, or policies. Online comments about the temporary transbay bus terminal are universally negative. Emeryville is not especially easy to get to by public transportation, either; you would need to take BART to MacArthur then the Emery Go-Round shuttle bus.
Speaking of BART, I have seen some people advise renting a bicycle locker, but this takes a lot of advance planning, and presumably, a local address for your application form, only to leave your bags half-exposed to the elements and at risk of theft. I don't recommend it.
Further down the list, you could rent a car, stick the bags in the trunk, and find someplace safe and cheap to park for the duration of your excursion. I extended a rental car for a day once precisely for this purpose in Hawaii, but have no recommendations for the Bay Area.