It is possible to run an engine without the exhaust. However, it's not advisable because it will make the engine crazily loud. Also, running an engine without exhaust can be dangerous to your health, as the fumes will enter the cabin and you would be inhaling all of those fumes and gases. So, if you're having problems with your vehicle's exhaust system, get it fixed as soon as possible. There is also huge a risk of fire, due to the exhaust combustion coming directly out from the cylinder head.