You would really have to consult with an attorney in your area of Ohio.  Some states have it codified - actually written in to the law - that a landlord can choose not to rent to a person as long as the reason is not based upon those categories that are protected from discrimination under the law: race, religion, gender, sexual orientation, etc.  Your category here is not one of those that is protected under the law.  And although it is unfortunate that the owner is not willing to hear you out with what the real story about those criminal matters may be, I think that he or she may be well within their rights to deny you an apartment.