Not sure what country you're in, but in the U.S., museums are non-profits. They aren't run by the government. Much of the staff is also volunteers. Actual civil servants tend to be more busy than less busy. The inefficiency of government, where it happens in public-facing jobs, is more to the sheer amount of bureaucracy and paperwork anyone needs to tackle do something, but busywork is still work. They're not being paid to sit around.
Very likely, things like manning an extremely low-traffic store or working night security at a place that never has any crime, anything where a warm body just needs to be present but whether work needs to be done entirely depends on whether anyone shows up or not, are your best bet.
Possibly babysitting. If you can get the kids to sleep or sit still watching television, you only need to be there and able to respond if necessary, not actively engaged through your entire shift.