Everyone goes straight to the apocalypse when these scenarios are contemplated. I think the DR650 is probably the best "get out of Pensacola when a cat 5 hurricane is approaching and everyone is going insane, lines are miles long for gas, and all the roads and freeways are packed with cars." You could ride off the side of the freeway in the grass, follow some railroad tracks, or take dirt roads, fields, etc to get out of Dodge. I bought the DR for completely different reasons, but unbeknownst to me I actually bought a pretty decent shtf vehicle. Of course, useless if your not single, if you have big pets, etc. But for a single dude looking to grab some clothes, a laptop, and some small valuables and get out of trouble, its perfect. Strap a few gas cans to the bike (with an IMS/Acerbis/Safari already on it), and you can get 3-500 miles away from trouble before you even need gas. Its simple, nothing to break or overheat, etc.
As far as total Apocalypse, I would take a bicycle over anything. A horse can be shot by someone, it needs food and water (like you do), it cannot be thrown over a fence, and it can be spooked. A bicycle is literally almost bulletproof. Anyways..