You can run containers on exozyme using Podman, which is basically Docker but better. Alternatively, you can use Distrobox if you would like an easier, automated way to create and manage containers. If you aren't familiar with containers, first read this intro.
Run containers when logged out
By default, containers won't run when logged out. This is due to users only having access to rootless podman.
To fix this, run loginctl enable-linger $USER. This makes systemd run your services (including podman) even if you logged out.
Usage
Read this Podman tutorial to learn how to use Podman.
There's a graphical web interface for Podman at exoportal.
To run docker-compose.yml files, use podman-compose.
Tips and tricks
To use Docker images from the Docker Hub with Podman, you need to prepend docker.io/ to the image name, for instance, docker.io/weblate/weblate instead of weblate/weblate.