For a very long while concern at disposal of radioactive waste has been psychological, emotional and political rather than scientific. Low level wastes have been stored in hospital storerooms for decades rather than hospital authorities taking the initiative and bearing the cost of transporting the material to somewhere more sensible. Taking up valuable hospital storage with radioactive waste may be foolish but it has also been safe.
Medium level waste does get stored in purpose built facilities. We have actually built these in Australia with the support of local communities but against entrenched irrational opposition of people whose vested interest is to maintain a "no storage is available " ideology.
High level waste does get stored safely at reactor sites all around the world with the current exception of Fukushima. At this location, storage is currently precarious but my expectation is that safety will improve by several orders of magnitude over the next few years. (I declare a personal involvement in this work.)
After about thirty years of temporary storage, the radiation level of high level waste is reduced sufficiently to allow specialised transport. This transport has been taking place safely for decades. There are purpose built high level waste facilities in a number of locations worldwide. France for example reprocesses high level waste and the US investigated a facility at Yucca Mountain and has actually constructed a facility in Eddy county< New Mexico.
The failure to build at Yucca Mountain is often cited by nuclear power opponents as a failure to address high level storage but as the above link shows, this is not the case.
The rhetoric goes on but the nuclear reality is that we are producing nuclear power at many locations around the world and overwhelmingly we are doing it in a safe and responsible manner.
