I've found some guides which say that this used to be complicated, but is now as simple as
sudo apt install obs-studio && sudo apt install v4l2loopback-dkms
This does indeed give me a "Start Virtual Camera" button on OBS Studio. When I click it, I am prompted for a password. And that's it. Nothing else happens. The button still says "Start Virtual Camera" (it should change to "Stop Virtual Camera"). And no virtual camera is detected by Firefox, Zoom, or Cheese.
Am I doing something wrong?