Your best chance of straightening this out in a timely manner would be through the use of an attorney. Proper documentation will needed in order to verify that you do not have an record and you will likely have difficulty acquiring said documentation on your own.
I agree with my colleagues, that you should definitely consult with a lawyer, and also find out what information it was that disqualified you. It could be someone with a similar name, it could be any of a number of reasons.
But what kind of lawyer - you may want an employment attorney - since you were turned down for the job. I generally advise my clients to go to Department of Justice and have your rap run. Get a copy. If you have lived your whole life in the same town you may be able to look up criminal cases on line - if not the courthouse has a way to search - and see if there is a case and/or conviction. Strangely enough the DA may help you if it turns out they can convict the person that used your name.