Nuclear war... like mdk said, it's like Russian Roulette. No winners, just survivors. I doubt that even a full-fledged nuclear war would be enough to wipe out the human race entirely. Probably be something like Toba at the very worst, I think.
That said, if... or if I'm to believe my pessimistic tendencies, when the nukes start flying... it's not gonna be all Let's Go Sunning on your little Pip-Boy radio while you're blowing up raiders and smashing a super mutant's skull in. It's gonna be hell on earth. Maybe not literally, if there's gonna be a nuclear winter. But if I survive, I'm gonna suffer. If you survive, you're gonna suffer. A combination of smarts, strength, and stamina might make things a bit easier, but it's still gonna be anything but easy. I'd consider myself lucky if I got vaporized in a nuclear blast. I'd probably still be lucky if I died not too long after the bombs have fallen. But to think about how drastically my life would change... think about it. No internet. No video games. No showers. No electricity. No groceries. All the shit we take for granted will be gone. Maybe not right away, but it'll be gone quicker than you'd think. And things aren't going to be as easy as breaking into your neighbour's home or a nearby gun store to take their guns. Guns and ammo are going to be as precious as anything else. More than likely, if you want a gun... you're gonna have to kill someone for that. Someone that could very well have been someone you knew and were friends with before the bombs dropped.
Fuck... if there's anything scarier than wondering what would happen to me in this sort of shit scenario, it's even scarier thinking about how sick and depraved I could probably end up becoming myself if I did survive. I already have enough issues that make it hard for me to get along with people. Even my own friends and family. The apocalypse would probably be the one thing that would make me cross the moral event horizon into some really horrifying shit that already bothers me enough right now as intrusive thoughts. It'd be a dog eat dog world, and if I survive... and become stronger as a result... I'd probably end up becoming the worst out of the worst. It's anything but pleasant to think about.