The problem

The app on web-01 logs connection refused when it calls the database. You are root on web-01 and go through the usual checks: names, addresses, listening ports, then reachability.

web-01 has one network interface, eth0, and a service of its own listening on TCP port 8080. The database runs on db-01, port 5432. For DNS you check example.com against the default resolver and against the public resolver 1.1.1.1. The nc installed is the OpenBSD netcat.