Try the IT industry - there's loads of this sort of thing.
In my line of work (IT in a smallish private healthcare organisation), my team operates our organisation's helpdesk 9-5 and responds to all types of calls (usually referred to as first/second/third-line). Whilst my real job is Manager of the dept, I'm the only staff member of the in-house team who is classed as being able to respond to third-line calls.
Outside of those hours we have
outsourced 24/7 cover for our helpdesk, which covers first-line inside the contract. The first-line guys at that company will log and triage all calls onto our helpdesk, closing any off they can, especially if it's something daft like no paper in a printer. If something bigger happens which needs to be resolved ASAP, they have a second-line available (who may also be in bed!) who can take the call and r
esolve it, billing us for any hours as necessary (the number of these calls a year are so few, it's cheaper not to have this service in the contract).
Should safety of life be an issue (something which can and does happen in healthcare scenarios), that will instantly be forwarded to the on-call manager for the relevant hospital/care home (if they weren't the one to report the issue), who can make a decision to call me. Effectively that places me on-call 24/7/365, although the number of calls for this sort of thing is maybe 1 or 2 a year. Do I get handsomely remunerated for this role? Noooope
That said, I am in a management position and it works well enough for me, and it isn't something I would expect of lower-grade staff in my team.