Huh, it just occurred to me that Stack Overflow would still have access to their own database and probably be able to run a copy of it locally.
If Stack Overflow went down, they'd still be able to check Stack Overflow to fix their issue.
Stack Exchange (and by extension, Stack Overflow) publishes all their data (anonymized) for download and offline usage as part of the Archive.org project. If you wanted, you too could have a copy of it locally to search while the site is down.