Most of the "half my house" answers are incomplete. Most of the time the problems is in the equipment that is in the utilities jurisdiction, but it certainly can be an issue with the load side of the meter, the wire to the panel, the connection to the main breaker, internal of the main breaker, or connection of the main breaker to the bussing.
You need to get your landlord to get somebody out to investigate the customer side of the service before some more damage results.
More explanation is needed about how the connection at the weatherhead is not utility jurisdiction.
Edit: You have a split bus "rule of six" panel that up to 6 "main" breakers were allowed to shut off all power. One of the six 2-pole breaker usually near the middle of the panel feeds all the small breakers in a sub-section of the panel. In your panel it is likely the 9,11 marked "main/lights". As long as that one is off none of the lower breakers will do anything. After turning off all 2-pole breakers turn only the 9,11 on and half your plugscircuits will come on, turn on the lower water heater breaker and weird stuff will start. You're missing a leg and the 120v circuits connected to the dead leg are feedingbeing fed reduced voltage through the water heater.
I would suspect that when 5,7 failed it also caused damage where it attaches to the bussing, so they added the bottom 2-pole breaker to the "lighting" sub-section and moved wires down to it.
Still, you need an electrician to find the faulty connection. I absolutely recommend only people with electrical training and flash retardant suit risk removing panel cover when a likely failed wire termination lug is possibly hot enough to come apart. Seriously.