Great, now come back to me with a plan to plant the trees required to offset carbon emissions. This is going to be challenging, because you'll need to reforest an area roughly the size of the continental United States to reduce atmospheric CO2 to levels from a century ago [1]. In order to actually do this, you'd need to desalinate massive amounts of water to feed trees in areas without vegetation. Generating the energy for desalination and pumping it inland would release massive amounts of carbon back into the atmosphere, necessitating more tress, necessitating more water, and so on. You'd probably have better chances sequestering carbon through an algae bloom in the oceans, but that too has drawbacks (namely, when the algae dies off it'll just release the carbon again).
By comparison we already have a nuclear waste facility built in the US, though it's use was cancelled by Congress (and it wouldn't make sense to put spent fuel there yet, since we would want it for reprocessing), and another site is under construction in Finland [3].
"Bury it underground, in an area with no aquifer" isn't hand wave. Storing nuclear waste really is that simple. The main concern for uranium is water contamination. It's actually a prevalent due to naturally occurring uranium [4]. Place the waste in an area with no aquifer and there is no risk of contamination even in the event that the containment vessels deteriorate.
Q: what is the long term storage plan for fossil fuel emissions?
A: Trees