I’d like to read Ed Bearss’ Indiana University thesis on Patrick Cleburne.
I’ve googled a bit but since he’s a pretty famous historian I’m not able to wade thru all of the results I get very well.
Call up the IU History Department and ask them. It may be in the university library somewhere, or the department should have a copy on file. I know people have gotten copies of mine this way.
If it is from 40 years ago, it may well be that the only copies in existence are those in the IU library (at the relevant campus) and retained by the author himself. You might be able to get it through inter-library loan (your chances will be better if you are a patron of a university research library, but asking at your local public library might work).
It also might be available from these people http://www.proquest.com/en-US/products/dissertations/ In the old days they would microfilm a lot of dissertations, and they may have digitized their microfilms by now. However, the best I way to access them would probably be, once again, through an inter-library loan service. It may be possible for individuals to deal with the company directly, however.
Your last resort might be to take a trip to Bloomington (or whichever is the relevant campus) and descend into the library stacks yourself.
It occurs to me, however, that if, as you say, the guy is a well known historian by now, he has probably published must of the content of his thesis, in some form, in regular books and/or scholarly articles by now. Those might be easier to access.
It’s 56 years old, not 40, so I’m not surprised that UMI (University Microfilms International) doesn’t seem to know about it. The IU libraries appear to have two copies, which ups the probability that they might be willing to lend one copy through inter-library loan. Talk to the folks at your local library. See the IU library catalog entry here.