Many reactor designs have an “Emergency Core Cooling System” consisting of a tank of water on the roof of the reactor building, that runs into the reactor to cool it in the even of a failure of the main cooling system.
A 1GW reactor generates 1GW of electricity. To do that, it generates around 2GW of heat (rather more for older designs, a bit less for newer ones, depending on the thermal efficiency of the system). In an emergency, the chain reaction can be shut down very quickly, but the radioactive fission products continue to generate heat – initially about 6.5% of what was being generated before, that is about 130MW. This diminishes over the following hours and days as the fission products decay.
The heat of vaporization of water is about 2,260 kJ/kg, and raising 1kg of water from an ambient 20°C to 100°C takes 335 kJ, so 130MW will boil 50kg of water a second. After about an hour this will be down to about 12kg/s, but diminishing more slowly – it will still be over 1kg/s after a week. How big is the tank of water on top of the reactor building? If it’s 500 tonnes, that means you’ve got to arrange a new source of cooling water within about six hours...and it’s Fukushima all over again if you don’t manage, for whatever reason. Not forgetting that this must be pure water – or at least, water with no solids in suspension or in solution, because any solids that are left behind when the water boils will obstruct the flow of coolant. Sea water, or even normal tap water, won’t do.
Actually, it’s worse than that, because emergency core cooling systems don’t work nearly as well as that. You can’t ensure that the coolant reaches all parts of the reactor in proportion to the amount of heat being generated in each particular part of the reactor, so some parts will be cooled more than is necessary, while others are insufficiently cooled – unless you use more water than the simplistic analysis suggests. Actually a great deal more. One of the big problems is the Leidenfrost effect: once the pressure is off the system, the water doesn’t get to touch the hottest places at all, with an insulating layer of steam between the water and the hot surface. (This was the subject of one of my big arguments with my professor in 1970, see Nuclear Engineering – a bit of personal history.)
You can adjust the figures for different size reactors, different thermal efficiencies, and different sizes of water tanks – but the broad picture remains the same.
(There are variations on this theme, with pumps delivering water from rivers or lakes – but the tank on the roof can be a “passive” design, which in principle should be more reliable. River or lake water is also liable to contain dissolved solids in most places, and intakes are liable to become clogged with weed, algae or debris.)