I don't understand why you want to use exim instead of postfix. In this file
included in RHEL 6 documentation it is said:
Exim has been removed from Red Hat Enterprise Linux 6. Postfix is the default and recommended MTA.
The document "Setting up exim" which you refer to date back December 8th, 2004. That's pretty old and it is about exim 4.2, not exim 4.8 that you downloaded
Anyhow, in both the aforementioned document and exim-4.80/src/EDITME you are told to first make a directory called Local in the distribution directory, then put a copy of exim-4.80/src/EDITME in it, renamed as Makefile, and edit this Makefile before running the "make" command.
Let's suppose you downloaded the exim-4.80.tar.gz in your home directory (I'll call it /home/gaurvrishi), the tar command will have created the directory /home/gaurvrishi/exim-4.80. Then do this:
cp /home/gaurvrishi/exim-4.80/src/EDITME /home/gaurvrishi/exim-4.80/Local/Makefile
Now you can edit the Makefile.
Anyhow I am not sure at all that exim-4.80 can be compiled easily in RHEL 6 and my advise still is to install postfix instead, following instructions given on Red Hat's web site.