@cicerone imposter,
The Nevada site is for the disposal of spent fuel from our commercial reactors.The right way to do this is to reprocess the fuel chemically, extracting the still fissionable Uranium, plutonium and Thorium (Thorium is not directly fissionable but it breeds fissionable nuclides itself) from the fuel, thereby reducing the waste stream and producing anouther round of fuel with nearly the energy producing potential of the original. Unfortunately we don't do that.
Some of the nuclides in nuclear waste have very long half lives, and they need to be isolated in a stable environment for a long time. Happily the volume of such wastes is relatively small, and it is a readily solvable engineering problem. The French store theirs in tunnels in the Alps. Yucca Mountain is an even better designed and more stable facility.
The nuclides in the weapons you were handling were mostly alpha emitters which can't even penetreate your skin. They were no external hazard at all, but I wouldn't recommend eating the stuff.