Why can't nuclear reactors power themselves?

boiling water nuclear reactor system

While I was transfixed with the unfolding of the Fukushima Daichi nuclear reactor tragedy in Japan following a massing earthquake and the ensuing tsunami I could not help myself but occasionally stop and ponder: "why can't nuclear reactors power themselves with electricity?" After all it was the lack of electricity needed to power the water pumps to cool down uranium fuel rods that is responsible for the runaway nuclear reaction and the resulting meltdown. From this, one thing is clear - nuclear reactors need electricity from the outside to power the cooling pumps otherwise the meltdown is practically inevitable.

But wait! Nuclear reactor's sole purpose of existence is to produce electricity. Heck, there are stand alone nuclear reactors out there that power individual submarines, carriers, whole cities, and of course my pocket watch... But no really why can't the damn thing power itself? Is this some sort of Catch 22 or just weird science???

While we may never know the answers to the above questions the good news is that newer 'safer' nuclear reactors with passive backup cooling systems are in development and construction as we speak. These next generation reactors will make use of natural forces like gravity and convection to cool down in the event of failure of pumps or external power.

nuclear reactor passive backup cooling system uses gravity and convection

[Are there safe nuclear reactors?]