I've seen in Spain that police drive around telling people to go home, and have also used drones with loudspeakers to fly over public places telling people to go home. Pharmacy and food shopping is allowed even in lockdown. Just today, one of the senior chief medical officers said children can still go out and play but as a family unit, not with vulnerable people, or adults.
As I understand, the government (yet) can't legally enforce a full-on quarantine, e.g. where it arrests people who go out (I could be wrong on that!), so it's shutting places where people are likely to spread it to discourage people going out. As a key worker, you'd go to work as normal I assume.
The dystopia is here!