The two separate tickets that you hold do not constitute a "single contract of carriage" and therefore QR does not have any obligation to protect / reaccommodate you. In most likelihood, in case you miss your connection, you will have to call AA and sort things out with them (beware: DHIA has quite good free wifi however roaming charges are obscenely expensive) - in this case though for AA you will be no show so please review the conditions of your ticket that regulate such cases, you may have to pay a hefty sum to be rebooked. I also see a different issue here - as you have a one-way JNB DOH ticket and Qatar is currently closed for non-nationals / non-residents, it is quite possible you will not be allowed to fly at all (especially if you have luggage to check in, as there is no guarantee that QR will be able to check it through to ORD, so that you can claim that you will remain in transit). To evade any surprises, I very strongly suggest that you call QR and discuss your circumstances - or, even better, if you can, go to JNB / QR local office and discuss things in person with the QR local rep. As of 1 June QR is scheduled to operate only 1 daily rotation on DOH ORD - if you miss that and provided that access restrictions to Qatar remain as they are currently you will have to spend the time until the next flight in the transit area of the airport (or alternatively, book a room in the transit hotel). As an alternative and depending on the rebooking conditions of your AA ticket you may potentially be rebooked on other OW carriers, if any such flights operate by the time you travel.

An obvious comment but just making sure: the fact that JNB DOH is a repatriation flight does not entitle you to any different treatment than as if it were a regular scheduled flight (unless your ticket conditions specify otherwise).