I don't think anyone is proposing continuous fuel (even fission reactors typically don't work like that - you have to shut them down to swap out the fuel rods periodically). The idea is that you inject (say) some deuterium into a blast chamber, it explodes like a miniature thermonuclear bomb, and you gather the explosive energy in vaguely the same way a internal combustion engine (like in cars) gathers energy from fuel-air bombs in its engine cyliders.
And we already know (roughly) how to build that, it's just that the version we know how to build is so ridiculously, impractically huge that you could fit a small city in its blast chamber... because it runs on actual, non-miniature, thermonuclear bombs. The difficulty is scaling that down to something that doesn't have a upfront cost measured in multiples of the world GDP before it starts generating power (in quantities measured in multiples of current global electrical consumption, and likely multiples of global energy consumption period).
The blast chamber that you chuck nuclear weapons in is not a good analogy, I suspect we don't know how to build such a thing that has a net positive energy yield when you factor in how much energy it takes to build a thermonuclear weapon and the terrible thermal efficiency such a contraption would have. Inertial confinement fusion is basically a more manageable version of this idea and we still haven't hit breakeven there. Magnetic confinement fusion is widely believed to be the more viable path to commercial fusion energy anyway. Right now fusion is largely a game of chasing margins to get things efficient enough that first the plasma is self sustaining (ignition) and then second yields a net positive of energy. ITER or its successor will hopefully achieve the first goal (ITER was originally supposed to but is no longer targeting full ignition, at least initially), then hitting the second goal is what will be needed to be practical as a power source.
Also, just to add to your point there are (basically) continuously fueled fission reactors, see e.g. CANDU as well as some more exotic actually continuously fueled designs that use liquid fuels. Continuously fueled designs would probably be more desirable if they weren't such a proliferation concern but that's a different story.
Wait, are you saying we know how to build a World Wonder that produces infinite energy, but we’re not doing it? What’ll we do if Ghandi builds it first?
Because we don't have enough energy demand to make it worthwhile. Even if we had enough money (and political will) to fund it, we'd be better off spending it on a space elevator, since we can actually use that enough to cover the costs.
And we already know (roughly) how to build that, it's just that the version we know how to build is so ridiculously, impractically huge that you could fit a small city in its blast chamber... because it runs on actual, non-miniature, thermonuclear bombs. The difficulty is scaling that down to something that doesn't have a upfront cost measured in multiples of the world GDP before it starts generating power (in quantities measured in multiples of current global electrical consumption, and likely multiples of global energy consumption period).