They can't simply store that ever growing volume of water forever. But it seems it should be possible to filter out the radioactive material somehow. It's this kind of thing that really argues well against nuclear fission as a power source.
But it seems it should be possible to filter out the radioactive material somehow.
I've heard that all the heavy isotopes have been removed insofar as that is possible. I don't have hard evidence of that, but that's what all the mainstream sources say. Your mileage may vary.
What they cannot remove by filtration or other conventional processes is tritium. Tritium is really water - heavy water - and they tell us that there is no way to separate water from water. That is why they have to dump it. I could be way wrong, but that's the story.
"The half-life of the unstable tritium nucleus is of 12.3 years, which is very short on the radioactive time-scale. This comparatively fast disappearance means that very little tritium can accumulate in any place." - www.radioactivity.eu.com site pages Tritium
The Fukushima disaster occured in 2011. I don't know if the half life information tells us anything about oceanic hazards.