Do not disturb mode doesn't turn off access. It is only meant to turn off sounds for certain things. If it is stopping your device from working completely, I would start to wonder if it is actually the phone's option or a virus, malware, etc.
Try turning off the device and then back on. It should work correctly if it is just the phone itself. If it doesn't, then try turning it off and start it up in "Safe Mode". While this mode gives limited access, it will also let you know (if it works correctly in it) that there is in fact something on the device causing the problem that you will then have to remove to get it working correctly. I would suggest uninstalling any apps recently added and also uninstalling any recent app updates. then...