If you want to buy a house in CA without making your address public to anyone who searches for your name, it will be... tricky, probably.
In CA, there are (at least) three public records that can provide a determined seeker with a copy of your address:
The title (ownership record) and deed of trust (mortgage record) are public records. Both documents will contain your name, your co-owners/co-borrowers' names, and the address of the property.
Your voter registration is not public, but the full list of all voters with contact information is regularly sent to eligible political and academic groups.
Buying a house privately means working out how to remove either your name or the property address from those public records.
Here are the tools (that I know about) that you can use to work on this problem.
In parallel to any of the other options on this page, you should probably be paying a service to actively send takedowns to all the companies that scrape public records and sell copies of the data. That includes phone numbers, mailing address history, ownership records, and anything else that can be scraped or bought.
I’ve personally used Abine’s DeleteMe, PrivacyDuck, Removaly, and Optery. For details see the appendix at the end. tl;dr, I’m using Optery today and it seems okay.
Privacy title, deed of trust, and voter record
Cost free, courtesy CA Secretary of State
Requirements must be (or be household member of) a victim of domestic violence, stalking, sexual assault, human trafficking, or elder/dependent abuse OR a local health officer, public health official, reproductive health care worker or volunteer
Process Apply through a victim advocacy organization approved by CA. In SF, options include La Casa de las Madres or WOMAN Inc.
Members of Safe At Home are given a mailing address administered by the state. All mail to that address will be forwarded to you at your true address.
By CA law, government agencies (like the county recorder's office that maintains public documents) are required to use your Safe At Home mailing address on both title and deed of trust, instead of the true address of the property.
If you are a member of Safe At Home, you are also automatically qualified for CA's Confidential Voter program, which will keep your home address out of the voter registration lists as well.
Privacy title only
Cost ~$3500 one time / ~$500? yearly