Coming in somewhere between 1 and 5 Mb in on-disk size (depending on the variant), BusyBox is a very good ingredient to craft space-efficient distributions.
BusyBox combines tiny versions of many common UNIX utilities into a single small executable. It provides replacements for most of the utilities you usually find in GNU fileutils, shellutils, etc. The utilities in BusyBox generally have fewer options than their full-featured GNU cousins; however, the options that are included provide the expected functionality and behave very much like their GNU counterparts. BusyBox provides a fairly complete environment for any small or embedded system.
%%LOGO%%
$ docker run -it --rm %%IMAGE%%
This will drop you into an sh shell to allow you to do what you want inside a BusyBox system.
Dockerfile for a binaryFROM %%IMAGE%%
COPY ./my-static-binary /my-static-binary
CMD ["/my-static-binary"]
This Dockerfile will allow you to create a minimal image for your statically compiled binary. You will have to compile the binary in some other place like another container. For a simpler alternative that's similarly tiny but easier to extend, see alpine.