None of the existing answers address the case of a container that exited (and can't be restarted) and/or doesn't have any shell installed (e.g. distroless ones). This one works as long has you have root access to the Docker host.
For a real manual inspection, find out the layer IDs first:
docker inspect my-container | jq '.[0].GraphDriver.Data'
In the output, you should see something like
"MergedDir": "/var/lib/docker/overlay2/03e8df748fab9526594cfdd0b6cf9f4b5160197e98fe580df0d36f19830308d9/merged"
Navigate into this folder (as root) to find the current visible state of the container filesystem.