Why the water heater is the landlord’s job
The landlord is generally responsible for repairing or replacing a broken water heater, because providing hot water and maintaining the plumbing are part of the landlord's duty to keep a rental livable.
A water heater that stops working, leaks, or cannot keep up is the landlord's to fix — the same way the landlord is responsible for the pipes and fixtures. A tenant is not expected to buy or install a water heater. A leaking water heater deserves extra attention, since it can also cause water damage and, over time, mold.
If your water heater has failed and you are left without hot water, that is a habitability problem the landlord must address, usually promptly. Report it in writing and note when it stopped working. If your landlord will not repair or replace a failed water heater, a free review can help you understand what to do next.
This is general information about California tenant rights, not legal advice. Every situation is different.