Take a look at docker export.

To quickly list the files in your container:

docker export CONTAINER|tar -t

To export:

docker export CONTAINER>snapshot.tar
docker export CONTAINER|tar x PATH-IN-CONTAINER

Or to look at a file:

docker export CONTAINER|tar x --to-stdout PATH-IN-CONTAINER
# e.g. 
docker export consul|tar x --to-stdout etc/profile

Docker 1.8 supports cp:

https://docs.docker.com/reference/commandline/cp/

Usage:  docker cp [options] CONTAINER:PATH LOCALPATH|-
        docker cp [options] LOCALPATH|- CONTAINER:PATH

update: you should ssh to your docker machine when you run this.

docker export does not export everything about the container — just the filesystem. So, when importing the dump back into a new docker image, additional flags need to be specified to recreate the context.

For example, if the original container was running fine because the Dockerfile that was used for creating its image had CMD ["/usr/bin/supervisord"] in it, then import your dump this way:

docker import \
--change 'CMD ["/usr/bin/supervisord"]' \
path/to/dump.tar imagename:tagname