Nuclear waste disposal is difficult because of the nature of the beast. This isn't a complex molecule(s), like oil, which can be digested by bacteria, we have single atoms which are Nature's Time Bombs. Thus, they can be very mobile in natural systems, until such time that they decide to decay into daughter products. During this process, the can emit high-energy particles, or various forms of radiation. It is called ionizing radiation because when it zooms through a living system, it damages complex molecules. It takes it's path, causes it's damage, and vanishes. Sometimes the daughter products are still radioactive, and can blow up again.
For most radionuclides potential damage is confined to a very small zone in living systems. We live with protein and DNA damage all the time, the body recognizes and eliminates the damage. We old men can take a lot of radiation, and it's probably good for us, up to some point, knocking out potential cancer. It can be dangerous in situations where the body cannot recognize the damage, such as the moment of conception for young women, or rapid growth with youngsters. And, of course, the poison is in the dose, so that a lot of it is going to fill you with toxic free radicals, and you will succumb to radiation sickness. If the particle decays outside your body, in most cases the ionizing particles or radiation cannot penetrate your skin. You do not become radioactive with exposure to radiation. When you have your chest xray, you take the damage, and 'Suck it up, Princess'.
-look up radiation sickness, radioactive decay, medical isotopes.
The slippery nature of these radioactive atoms provides a challenge for confinement. Luckily, the biggest, nastiest atoms are slow and awkward, and get stuck in anything. We just have to stop people from eating, drinking or breathing them. You don't want a speck of Plutonium stuck in your gut!
Over the billions of years of the Earth's geological existence, it has been exposed to more loose radioactive contamination that than we puny humans could ever achieve. Radioactivity drives the Earth, without it we'd be a dusty speck. Fortunately, all the nasty radioactive isotopes are promiscuous, in that they'll bond with anything. Thus, we have it all locked up in rock. We don't eat rock.
-look up adsorption, sediments, oceanic black smokers.