Can you run an engine w/o headers ?

I think you need some sort of exhaust ,My father once told me of a time when someone he knew ran an engine with just a very short straight pipe off each cylinder the engine ran fine but when they shut it down the cold air running into those short pipes damaged the hot exhaust valves. Keep in mind this was probably in the 1950's .I'm sure materials have changed since then .
But I always remember that story. I mean why risk it ? JMO