Chernobyl definitely had concerns with groundwater contamination. There were significant amounts of strontium within the exclusion zone, and the core had melted down into the earth, although I don't think it reached the water table. However, from everything I can find, the migration through the soil tended to be extremely slow, making it a fairly local phenomenon. In the end, cleanup efforts were abandoned since they had low impact on human health or radioactivity outside of the exclusion zone, and were worse for the environment than simply leaving things.

Unfortunately, I no longer have access to the journal subscriptions, so I don't have the full text of this, but I think this is the paper you want to read for an overview: http://onlinelibrary.wiley.com/doi/10.1029/96WR03963/abstrac...

The conditions are very different at Fukushima, mainly due to the proximity to the ocean, so I can't really give an educated summary of the challenges with that. One thing that you have to keep in mind, though, is that water is an excellent radiation absorber, and it's actually safe to go diving in active reactor pools as long as you're more than a couple of meters away from the core. I'd expect that the radiation in the ocean would disperse quickly and harmlessly.

https://what-if.xkcd.com/29/

Also, the region around Chernobyl isn't exactly known for the frequency of major natural disasters like earthquakes, tsunami, and typhoons.