What could cause my old gas furnace to keep cutting out?
I would check your ignition sensor or your flame sensor. There are also heat switches that "melt" and shut off to stop the furnace from catching fire. I had a similar problem when I bought my house that was a foreclosure. The furnace sat for a while and the sensor got covered in dust and burnt out when I turned the heat on. The best way to figure out what's up is to remove the cover and look for some blinking lights. It will depend on the furnace type. These lights will give you a code that will help you figure out what the real problem is. So 
Google your furnace model and how to find the code 
then 
Google your furnace model and the code to get an idea of the problem