Storing hazardous material in a subterranean formation

A method for storing nuclear waste hazardous material, comprising:

• moving a storage container through an entry of a wellbore that extends into a terranean surface, the entry at least proximate the terranean surface, the storage container comprising an inner cavity that encloses nuclear waste hazardous material;
• moving the storage container through the wellbore that comprises a substantially vertical portion, a transition portion, and a substantially horizontal portion, the storage container sized to fit from the wellbore entry through the substantially vertical, the transition, and the substantially horizontal portions of the wellbore;
• moving the storage container into a storage area that is coupled to the substantially horizontal portion of the well bore, the storage area located within or below an impermeable, hydrocarbon-bearing shale formation and vertically isolated, by the hydrocarbon-bearing shale formation, from a subterranean zone that comprises mobile water, the shale formation comprising a diffusion barrier to a radioactive gas isotope of the nuclear waste hazardous material;
• and
• forming a seal in the wellbore that isolates the storage portion of the wellbore from the entry of the wellbore.

A spent nuclear fuel storage system, comprising:

• a directional wellbore formed from a terranean surface, through a first subterranean layer, and into a second subterranean layer deeper than the first subterranean layer, the first subterranean layer comprising a rock formation that includes a source of mobile water, the second subterranean layer comprising an impervious, hydrocarbon bearing shale formation, the shale formation fluidly isolating a portion of the directional wellbore formed within the shale formation from the first subterranean layer, the shale formation comprising a diffusion barrier to a radioactive gas isotope of the nuclear waste hazardous material;
• a container configured to be moved through the directional wellbore into the portion of the directional wellbore formed within the shale formation, the container comprising a volume enclosed by a housing configured to store a plurality of spent nuclear fuel pellets;
• and
• a plug set in the directional wellbore between the portion of the directional wellbore formed within the shale formation and the terranean surface.