Hot water is the landlord’s job
Yes. California law requires rentals to have both hot and cold running water, so keeping the hot water working is the landlord's responsibility. A water heater that stops working, or plumbing that cannot deliver hot water, is a habitability problem.
Going without hot water affects basic daily needs — bathing, cleaning, cooking, and staying healthy. It becomes a legal issue especially when it continues after you have reported it, because landlords are expected to make repairs within a reasonable time. A complete loss of hot water should be treated as urgent, not something that waits weeks.
If your landlord will not act, you may have options such as requesting the repair in writing, contacting a local housing or code inspector, or pursuing a claim. Keeping copies of your requests and noting the dates the hot water was out will help show what happened. If this is your situation, a free review can help you understand where you stand.
This is general information about California tenant rights, not legal advice. Every situation is different.