Libraries would be the _one_ chance that humanity would have to reboot from a dark age.
Imagine the power went out. For a month. Or a year. E.g. we are very vulnerable to a massive solar flare like the Carrington event in 1859 (http://gizmodo.com/what-would-happen-if-a-massive-solar-stor...), which could knock out power for months or years at a time.
We'd need a way to maintain society without organized computer networks, at least in those areas. Given how interdependent we are, one really wonders if society itself would not collapse.
I know, it's crazy. But imagine you woke up to a world without any computers or computer networks. Maybe that world is 1 year after such an event, or 20. How would you reboot society, or the very least, find and learn what you needed to learn? The answer would be a _library_.
Books will survive. At least, until we digitize and upload all of them, then throw them away.
Apparently we have already thrown away the paper copy of the index. This is extremely dangerous.