They can use an online service in the middle of the night, but as a citizen, I'd be upset if I learned my public library were paging someone in the middle of the night if it weren't working. That's incredibly inappropriate. It can wait.
Like I said in another comment, computers generally don't just randomly stop working, but no, a library worker doesn't need their computer at 9am so badly that they need an on-call IT worker (and presumably some kind of middle-of-the-night monitoring) to ensure it before they get in. Patrons don't have some incredibly urgent need to borrow a book such that we need 5 nines of uptime here. Or any SLA at all. It's fixed when it's fixed.
This attitude is how you waste a ton of money overcomplicating everything while making working conditions terrible for literally no reason. We should not "take seriously" things which are not serious just for fun. Nothing bad happens if the library computer is down for a couple hours beyond maybe some mild disappointment, or perhaps manually writing the data on cards like we did 30 years ago for a bit.
Hell, most banks in the US are "closed" for maintenance online for several hours every weekend (or at least they were as of a couple years ago) and no one cares. It's fine.
I work on systems that are in the critical path of 24/7 manufacturing lines, so I'm familiar with things that actually do rapidly cause problems if there's an outage. But that's not every computer system in the world.
And if they have people across the globe, then problem solved; they don't need a 3am on-call.