The gov workers aren’t getting paid (the ones I know, anyway). They are supposed to go in to lock offices, turn in cellphones etc, for 4 hours then leave.
Some, of course, are still working. Obviously “essential” employees are still on duty (people whose jobs immediately affect safety, national security, and the like), plus there’s a lot of government that doesn’t fall under the spending in the budget process — anything funded by user fees, for example, doesn’t need a Congressional budget to keep in operation.