How long in ohio does a contractor have to finish a job after his contract expires?

When you say that his contract "expires" on March 21, I assume that you mean that the work was to be finished by that time.  I also assume that you have a written contract that spells out your remedies and that this shed is on your residential property. If you don't have a written contract, the contractor has other problems as that by itself if a violation of the Ohio Consumer Sales Practices Act "OCSPA".  You should talk with a qualified attorney.  Have them read your contract and advise you of your rights under the contract,  the "OCSPA" and other Ohio statutes. If the contracts says that he is supposed to be finished by a certain date, he doesn't have more time, unless he was delayed beyond his control.  You have to read the contract to see how that works.  The fact that he didn't pull permits may or may not be an issue in your community as some do not require permits for that kind of work, especially if there are no utilities serving the shed.  If your community requires that contractors register with the community and are "licensed" they may also have required that he post a bond.  That may be of help to you. A letter from a lawyer may be all you need to get him to finish the work or make some reasonable settlement with you.