I think you have a misunderstanding here, minimal and eclipse don't mix. There is nothing minimal about a full featured IDE written in Java like eclipse. In order for that to run you will need at least:
- A java engine
- A GUI, at the very least a simple Xorg but you would probably want to have a Window Manager or Desktop Environment as well.
As soon as you have those, your system is no longer minimalist. If you want a minimal system that can compile and execute C code, all you need is a kernel, a shell, a simple text editor to write the code and the core libraries needed to compile and run C programs. You cannot ask for minimal and expect to be running a full featured IDE.
That said, any Linux at all will be able to compile and run C programs as long as the necessary libraries and compiler are installed. Just pick a Linux that sounds good for you and try it. Have a look at http://distrowatch.com and check out Damn Small Linux for a tiny Linux distro.
If you want a normal distro that can connect to the internet and has a GUI (DSL has both these things mind you) and some basic configuration tools etc, go for Debian or arch or gentoo. Debian might have some programs installed by default that you don't like but it is very easy to remove them. See here for help on creating a minimal Debian. Arch and gentoo make it easy for you to build a DIY system and you can choose what extra functionality you want to include. However, they are not newbie friendly and if you are not already comfortable with *nix I would recommend starting elsewhere.