I am assuming since you emphasized hot water is out, that cold water flow is OK. If not, changes the picture and likely problem (assuming your water supply from outdoors has not failed - check with neighbors) is a bad pressure regulator or backflow preventer.
Hot water failure only:
If no hot water out of ANY faucets (try all in house as you likely have several branches in the water system), then clearly issue is almost certainly at water heater or close to it. If some have hot water but some not, then you have a blockage in only one branch and water heater is fine (unless you have two water heaters, one on each branch).
1) if some or all pipes only blocked - corrosion blockage - would be extremely rare UNLESS your flow rate of hot water had been steadily decreasing with time, and finally blocked off - commonly with iron algae in galvanized metal piping, otherwise very rarely in other metal pipes. Almost unheard of in plastic unless you have really cruddy iron or manganese rish water - like almost undrinkable. Also, VERY rare - an inadequately supported PEX (flexible tubing water line, if you have that) droops and kinks, cutting off flow.
2) All pipes out - if you have a constant hot water circulation pump that cirulates the hot water so it is always hot at the faucets, failure of the pump can, with some types of pumps and backflow preventer setups, cause near or total flow failure in the hot water system. Usually not total blockage, though. Of course, if you have separate pumps on different branches (very rare), would affect only that branch.
3) probably most common cause, and would be only if ALL hot water is out, is blockage of the inlet or outlet tube from the water heater. Rarely the plastic cold water filler tube in the bottom of the tank breaks (there have been a few brands over the past couple of decades where that was a problem, but most would have aged out and been replaced by now) so a piece floats up and plugs the outlet pipe, though rare for it to be a total blockage. Also, the outlet pipe can become blocked with corrosion - again, like with the pipes, would be preceeded by decreasing hot water flow over time. If the corrosion is in the heater itself it can be rodded out but probably means it is time to replace it because that fitting is likely to corrode out and leak soon. If in the piping above the heater then that can be replaced easily.
4) Rare, but an expansion tank installed with a pressure regulating valve on it (which it normally should not have been) can shut off when the control valve (mounted right on end of tank, which typically looks like a soccer ball sized metal can with rounded ends, but can also be a large galvanized tank several feet long on the wall or suspended in joists) corroded or fails and can block off flow to the water heater, which then means no flow from the heater either because it is getting no inflow.
5) Equally rare, but if the water heater supply line has a pressure regulator or backflow preventer only on it (as opposed to on the whole house system) that has failed, then can shut off the flow to the heater. In both this case and the above one, you would NOT have had significant pressure at the pressure relief valve where you checked the temperature - so if you have pressure there (steady flow when opened, more than just a short spurt) then incoming pressure control valve/regulator is not the problem.
6) If an expansion tank was installed in the hot water line after the water heater instead of the cold water line where it belongs, if its control valve (which it should not have in that application anyway) gets plugged then that can stop hot water flow.
Do the above tests to try to isolate it if you can - but either way you will be needing a Plumber unless you are pretty home handy. No way to estimate costs - minimum service charge of typcially $75-150 certainly, but depending on issue could run up to $1000 rangeif the water heater needs replacement due to heavy corrosion.