Some brave soul at Stack Overflow will have to physically go into the datacenter, roll up a cart with a keyboard, monitor and printer and start printing off a bunch of Networking answers.
The StackOverflow datacenter is famously tiny - like, 10 Windows servers. So even if the rest of the internet goes down hopefully they stay up. They might have to rebuild the internet out from their server room though.
Large portions of stackoverflow are archived and available for download with the Kiwix app (typically used for offline Wikipedia). Same with the arch wiki. Might keep it around for a rainy day.