The internet itself exists because the US military was seeking a way to maintain communications in the event that a major city was destroyed with an atomic bomb, causing a disruption in telephone communications.
Why, oh why do people keep trotting out this tired old myth?
The ARPANet wasn't created to survive a nuclear holocaust. Hey geniuses, it used common (though pricey and high speed) telco circuits - the same as carried telephone communications. They weren't hardened or anything like that. Explain to me how they'd stay put when everything else went kablooie?
The original purpose of the ARPANet was to allow resource sharing between research centers with computing resources that were being funded by and/or involved in defense level research. Even after the first dozen-odd IMPs (routers of their day, and amazingly only refrigerator sized, compared to the behemoths that they interconnected), they weren't even hardened.
Ironically, it would be over 20 years from the inception of the ARPANet that there would be a sufficiently large number of nodes and more imporantly links to give the Internet the level of robustness that might give it a reasonable chance of surviving an all out nuclear attack, the kind that people continually champion as its original raison d'être.
Anyone who's interested in learning more should really read the excellent book, _Where Wizards Stay Up Late_.