Exim doesn't seem to be recognizing my operating system type correctly, and so is failing to build.

Run the command scripts/os-type -generic. The output should be one of the known OS types, and should correspond to your operating system. You can see which OS are supported by obeying ls OS/Makefile-* and looking at the file name suffixes. If there is a discrepancy, it means that the script is failing to interpret the output from the uname command correctly, or that the output is wrong. Meanwhile, you can build Exim by obeying

EXIM_OSTYPE=xxxx make

instead of just make, provided you are running a Bourne-compatible shell, or otherwise by setting <small>EXIM_OSTYPE</small> correctly in your environment. It is probably best to start again from a clean distribution, to avoid any wreckage left over from the failed attempt.