The issue

systemd-resolved is used over D-Bus to resolve DNS queries made by Home Assistant, Supervisor, and add-ons. Without it, DNS will not work correctly in your installation.

The solution

If you see a message about an issue with D-Bus, resolve that first.

If the systemd-resolved service is not running or disabled, enable and start it.

If that does not help, reboot your operating system.

As a last resort, you need to reinstall the host running the Supervisor with one of the supported operating systems, see instructions here.