You're getting downvoted which is too bad because you're talking about the right plan for the wrong reasons.

Google for the term bioaccumulation. Your body, for example, has serious problems with bioaccumulating radioactive iodine and strontium and a couple other elements. The Japanese plan was to plant fungi that looooove eating cesium and accumulating it in their fungus flesh (well, maybe it wasn't cesium, but you get the idea). Then you scrape up the radioactive, dying fungus into waste barrels and plant some more, and eventually all the bioaccumulative element is in barrels instead of dirt.

There are of course minor engineering issues like how the heck you gather the bioaccumulated plant matter without using humans who will contaminate themselves and die and/or make a bigger mess than just leaving it alone.

There is also a little engineering problem with scale... say you had a major DDT spill which thins poultry eggs killing off wild birds. Well, you could put an industrial poultry fryer farm over the DDT spill and bury the DDT contaminated eggs in the bottom of the ocean, or maybe incinerate them or something to provide cogeneration heat and electricity to run the plant... yeah... it would just take a trillion birds a billion years to clean up a major spill. Other than that minor scaling engineering problem...

The fungi can't break down radiation, no known method can do that(1), and probably not even collect it. The amount of radioactivity is unchanged regardless of the presence of these fungi. They only use the radiation that is there as an energy source.

(1) Bombarding with neutrons or high energy charged particles may change the amount of radioactivity. Also, positron-emitting nucleotides can have slightly faster halflife at very low temperatures. None of these methods are available for the fungi.

Melanin has about a 30% absorption rate of ionizing radiation in the gamma spectrum, and remains stable (not oxidized in the traditional rate) while producing a current. The fungus is able to produce an acidic fuel source from the current that it feeds off of. The fungus is definitely breaking down radiation, incredibly, but it's mechanism is very poorly understood.

The fungus is (apparently) absorbing the radiation and using it as an energy source. But it is not making the radioactive atoms decay faster. So the time we have to wait until the area is safe and has a low radioactivity level doesn’t change.

True, but a layer of fungus would presumably act as an insulator, as it's absorbing part of the radiation? Perhaps it would be easier/cheaper/prettier to cover the sarcophagus in the fungus than in more concrete? Depending, of course, on a) how much it actually absorbs and b) how thick you can pile the stuff?

Nah, when gamma photon is absorbed it doesn't matter whether its energy will be used for fungal metabolism or converted to heat right away. And in case of gamma absorption rate mostly depends on mass density, so fungi will be likely as good as water. Concrete not only absorbs more radiation for the same thickness but is far more stable than anything alive.

what about the toxicity of the water as a result of being exposed to radiation, versus the toxicity of the fungi?

I'm not a nuclear scientist, but I can't think of any way pure water may become "toxic" due to exposure to radiation (instead of dissolving radioactive materials, which would definitely be a problem). Maybe excessive neutron radiation could build up tritium, but that already sounds like a stretch. (Most radioactive elements don't emit neutron spontaneously.)

Whoops, you're totally right! I misread the comment I responded to. Melanin is only useful when radiation is emitted. It can't coerce decay to happen more quickly.