I thought so too until not that long ago, but it turns out that it's a solved problem.
First, burying it in a geologically stable area is a technically perfectly acceptable solution. Only water soluble byproducts get transported out with the groundwater, and only at the rate at which the glass the waste is turned into can dissolve. So how quickly does glass dissolve? The short answer is "rather slowly". The long-lived byproducts travel much more slowly through the ground because they bind to the rock, then get dissolved again all the time. The scientists know how quickly each substance will travel away from e.g. Yucca Mountain, and they have all decayed before getting into the closest river. This solution is scientifically acceptable, but politically unacceptable simply because nobody wants to have the waste close to them even if it can't get out.
See chapter 11 of "The nuclear energy option" for calculations.
Second, this is one of the reasons for building
fast reactors like the
GE Hitachi S-PRISM ASAP. Fast reactors can burn stockpiled weapons material, normal fuel, natural uranium, depleted uranium and even spent fuel, and they can burn it until there is nothing left but short-lived byproducts, extracting very close to 100% of the fission energy. For comparison, LWRs can only extract less than one percent of the energy content of the mined uranium. The high efficiency reduces the waste volume by a factor of at least 20 (more than 100 if you consider depleted uranium a waste), and the waste will have decayed to radiation levels below that of the uranium ore in just three hundred years. Safe geological storage of something that goes away by itself in just 300 years is trivial.
For comparison, mercury from coal plants does not decay. It lasts forever, and we can only hope that it will in time form a chemically stable compound that will stay buried.
Fast reactors will need a lot of time to burn all the spent fuel we have already produced. On the other hand, each new fast reactor can take a full load of spent fuel as its initial core, so building them can remove a portion of the spent fuel that is currently being stored. But I think the psychological aspect of it might be as important - what used to be problematic nuclear waste has suddenly become energy for future generations. Also, there is no need to store spent fuel for millions of years anymore, we can turn all of it into energy in a few hundred years.