No lease, same protections
Yes. California's habitability protections apply to all residential tenants — whether you have a written lease, a spoken agreement, or a month-to-month arrangement. The implied warranty of habitability is automatic and does not depend on paperwork.
This means that even if you pay rent without any signed documents, your landlord still must keep the home safe and livable and make necessary repairs. A landlord cannot use the lack of a written lease as an excuse to ignore serious problems. Your other core protections generally apply too, including proper notice before entry and protection from illegal, "self-help" evictions like lockouts or utility shutoffs.
Having things in writing can still help you prove what happened, so it is smart to report problems by text or email and keep copies. But the absence of a lease does not lower the standard your home must meet. If a landlord is treating you as if you have fewer rights because there is no written lease, that is worth questioning.
This is general information about California tenant rights, not legal advice. Every situation is different.