Why is storage of spent nuclear fuel dangerous? Just what the title states; there's a good deal of noise made about transport, and storage of spent nuclear fuel. Why all the hullabaloo when the fuel is all spent?
 A: Mart's answer gets to some of the problem (e.g. ensuring that storage remains stable for the period of decay) but I believe the other answers here are a bit off base.
Nuclear fuel is determined to be "spent" when the nuclear engineers determine it is no longer economically warranted to continue using it.  On a physical sense, the fuel could certainly be used for much longer.  Besides, it depends on what kind of reactor you are talking about as to how long the fuel is used and what the composition of the spent fuel is.  The plutonium that is such a dangerous part of most used fuel is actually a major contributor to the energy output of a CANDU (Canadian heavy water) reactor toward the end of the useful fuel life.
As for composition of waste, the majority is composed of rather inert U238 (~91%) that did not transmute through neutron capture or fission.  This material is part of the original fuel composition and is not harmful.  A small percentage (~1%) consists of the remaining U235 that did not transmute or fission.  About the same amount is plutonium that results from neutron capture by U238 and subsequent decay.  Depending on reactor operation, about 4% is daughter products and the rest is actinides and activation products.
The half-lives of these isotopes varies significantly but it is a convenient fact that the more radioactive a material is, the shorter its halflife and consequently the shorter time before it is "safe."  There are many graphs (e.g. see second graph) out there of decay times for spent fuel, but as I said above, the exact time for decay depends on a lot of things like the original composition of the fuel, what kind of reactor was used, and the final processing of the used fuel.
Several options for processing spent fuel exist including recycling it to retrieve the useable uranium and plutonium.  Doing so reduces the waste volume considerably but also necessitates the development of separations technologies and the handling of concentrated waste.  It is also possible to place the waste in special reactors that are dedicated to "burning" the waste with high neutron flux; even still, there will always be some waste.  Waste that is slated for disposal is often vitrified, that is, it is mixed with borated glass.  These glass logs are put into steel containers and then stored in concrete.
Whatever is done with the waste, we must be confident in the stability of the storage for at least several hundred years (though thousands of years in some cases).  A great deal of research continues on this subject.  That being said, the absolute amount of waste is quite small.  I've read various numbers, but for order of magnitude we are talking about one football field 20 feet deep of waste for all of the nuclear reactors in the United States for the last 60 years.  That is a lot of bad stuff, but in comparison, it seems quite manageable.  If we recycled the fuel, that would reduce to about 6 inches of waste spread over one foodball field.  Coal plants, in general produce on the order of 10,000 times more waste by volume which contains more radioactive material in absolute terms than nuclear plant waste.
A: Here are the common engineering safety concerns regarding spent fuel:


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*Radioisotope source - in its normal form it is a danger to anyone next to it from penetrating radiation including gammas and neutrons, and there is a much much greater danger that should the fuel be torn apart it releases radioactive gases that emit radiation.  Even worse is the possibility that the entire thing gets dissolved into a water supply (like a Yucca Mtn concern).

*Heat production leads to danger of melting - there wouldn't be a strong concern about physical damage to the fuel if not for the fact that the fuel produces heat from those decaying isotopes.  This means that for a category of spent fuel (recently taken out of the core), active cooling is necessary or else it will just melt down on its own and create a small nuclear disaster.

*Potential to go critical - as has been pointed out here, the spent fuel only lack sufficient reactivity (from U-235 mostly) to go critical in reactor conditions.  However, reactor conditions are very different from ordinary life, and the doppler temperature coefficient of reactivity makes it less critical inside the reactor because it is higher temperature.  Our world is lower temperature, so a similar conglomeration (to a reactor) of assemblies submersed in ordinary water could create an active nuclear reactor.  This would be very bad.  In practice however, these assemblies are almost always coupled with absorber materials such that even very extreme conditions would not make them critical again.

*Proliferation and now terrorism concerns - Spent fuel can be recycled to use again, but the downside to this is that people can get access to somewhat usable nuclear materials through spent fuel.  The actual reprocessing process is very difficult but the difficulty depends on the safety and safeguards you employ - things a rouge regime may not be very concerned with.  The Plutonium, in particular, is very potent and can be isolated unlike the Uranium-235, although it would be suboptimal to use in a nuclear weapon, a crude weapon with reactor Plutonium is still theoretically possible.  Also, we are more worried about terrorism today and as I've pointed out, the material within the fuel is very dangerous when the physical integrity is not longer maintained.  There is some logic to say that this fuel is "self protected" because you can't pick it up and take it away on your own, because the radiation would kill you.  But concerns of a variety of types of attacks exist, since even bad people can be creative.

A: For a vanilla nuclear reactor (no MOX, breeder, etc.) the fuel (U-235) is only about 4% of the fuel rod itself. The problem with this, however, is that a "spent" fuel rod still has  U-235 (more than the natural occurring level of 0.71%) as well as U-238, Po-239, etc. Being "spent" just means that there isnt enough U-235 around to keep the chain reaction going.
All these materials in the fuel rods (as mentioned above) have thousands to 10000's years half life, so until they are "safe" to handle there will be nobody around to take care of it. 
The material that is enchasing these rods also becomes radioactive by the all the neutrons floating around and hitting the material. 
A: Well its not really all spent in the chemical fuel sense.  The leftovers from nuclear fission are themselves still radioactive, not usefully radioactive for energy generation.
These products will continue to be dangerously radioactive for 1000's of years and so the problem is to store dangerous materials for longer than modern civilisation has been on the planet.
A: As mentioned in the other answers, there's still radioactive material in the spent fuel. Any containment will have to deal with the heat from said radioactive material.
Also, radioactivity can change the quality of container-materials - metals can become brittle, glass might crack (see here: http://jol.liljenzin.se/KAPITEL/CH07NY3.PDF )
So you need a containment that stands up to radiation in addition to all other environmental stresses over a geological timeframe, while still beeing able to dissipate the rest heat.
