For docker, you have to create /etc/docker/daemon.json and set:

{
  "data-root": "/docker"
}

You can move your /var/lib/docker while the daemon is stopped. I had no issues when I did the move.

~~I don't know how to change this for podman.~~

LE: https://ahelpme.com/software/podman/change-the-location-of-container-storage-in-podman-with-selinux-enabled/

I would like to know this as well.

So far, it seems it's a bit easier to do with Podman / "standard OCI containers" because they're rootless and get stored in my home directory. But the solution I keep seeing is to move the directory and then symlink or mount bind the folder. I do the latter so that podman continues to work when that external drive isn't connected.

This does actually work, but I really don't like it. Why isn't there a way to store a container entirely in a specific location and then run straight from that location?

The alternative is to provide Podman a custom storage.conf for a specific location. But that too is a "permanent" change. I would love to know a cleaner solution to portable containers!