Will a theft offense turn up on federal background check and what are my options for getting this off the record of my former employer?

A few months ago, I was an employee at major retail store and stole $80 worth of merchandise over several weeks. I'm struggling financially and I needed the item to keep warm and when the loss prevention officer caught me I told them this circumstance and they sympathized (At least it seemed like they sympathized). I paid them the $80 back within one week and was very compliant with their questions. Now that I am thinking about it though, it may have been closer to $150 that I stole. I don't think it was that high though, but we all settled on $80 worth of merchandise. No police were brought in. I never had a court date, never stood in front of a judge. The only thing that happened was I talking to the loss prevention officers in the back room and then paying the store back for this loss. I am about to apply for a certain job at the US department of health. The supervisor told me that if they were contacted, they could only disclose that I was fired for "policy violation reasons" but nothing really prevents my supervisor for telling the truth. I know they keep my information in a database and federal background checks are extensive. Can they still prosecute? Will I be unable to get this job overall?