Do Docker containers take up space?
Yes, docker containers when they are running can take up some space as per your workload and application but as mentioned in the other answers - containers are built over layered filesystem therefore only extra space they consume is from some file/object they create/download during runtime which is not the part of base image like some debug log files, etc.
For ex. - When you run 100 containers of a base image of size 1GB, docker will not consume 100 GB of disk space as it leverages layered filesystem and top immutable layer is shared across all containers of same image.
When you stop and remove (yes!, remove explicitly or use --rm flag with docker run command) a container then it's space is reclaimed.
To check space used by containers (and other objects), you can use:
$ docker system df and for detailed info. use verbose flag $ docker system df -v
Initially when you start a container the SIZE column will show 0B but as soon as you start writing to the disk SIZE will change.
EDIT: added more info. as suggested by @Benjamin in comments