They aren't super common (like, each town has basically one), but here in Japan there are pediatric clinics with "sick daycare" where you can drop your sick chi

They aren't super common (like, each town has basically one), but here in Japan there are pediatric clinics with "sick daycare" where you can drop your sick child off in the morning to be looked after all day while you go to work. And they are (basically) free.