Yes, there are cameras that will work in extreme weather — at least on the more realistic side of each situation. There are many ruggedized cameras, almost any of which will cover some degree of each item on your whole list — although none will take being actually hit by lightning. A ruggedized camera can handle going to the beach (although you should be careful to get sand out of any moving parts after), but a sandstorm is a tall order and you would want additional protection, if you're really out in it. Likewise, these will handle snorkeling, but if you're diving down where there is significant pressure, you probably need additional precautions.

For nuclear disaster, assuming your own safety is either adequately dealt with or that you are past caring, no off-the-shelf camera will help. Radiation will fog film and can destroy digital sensors (and flip bits on memory cards). When photographer Igor Kostin illegally went into the Chernobyl site days after the explosion, he wore a lead suit and protected his camera gear with lead boxes — something similar would be recommended were you in a similar situation. Years later, a Chernobyl tour site notes that putting camera equipment on the ground is forbidden as it risks contamination, but doesn't offer further words of warning on protecting camera gear; presumably the tour is short enough and stays far enough away from high radiation areas as to minimize the risk.

For the more realistic scenarios, though, a camera feature search at Digital Photography Review turns up three dozen shockproof cameras, and I believe all of these results happen to be waterproof as well — which also means sealed against dust. You can narrow down the search from there.