Does docker-in-docker use memory or disk?
It uses disk.
A thorough explanation of how and why docker-in-docker works is in this article.
I originally asked this question with Teamcity's agent in mind, but I wanted to generalize the question. It seems that it is the industry standard never actually run docker-in-docker because it can cause data corruption, and most use cases can be solved with a docker-to-docker solution (explained below). Nevertheless, the nearly-docker-in-docker implementations are still referred to as docker-in-docker in some CI documentation even if it is not a true docker-in-docker solution.
The docker-to-docker workaround, generally, is to expose the host docker daemon to the container in a volume mount, i.e. docker run -v /var/run/docker.sock:/var/run/docker.sock ... And if you expose the daemon to a container, or expose the dind image to another container, in both cases the docker engine is running at the host level or as a container in the first level of docker, meaning that it uses disk.
Hope this clarifies.