My Dad went to Sears & Roebuck to get a replacement battery for the hat. The battery was a 22.5 volt/1.5 volt "Hearing aid battery". He asked for it by number and when the salesman came back from the stockroom he screamed at the top of his lungs "WE DON"T HAVE ANY!!!
Since the tubes are outside the hat (as shown) the tube heat would be dissipated externally. However I note from the Wikipedia article that it only had a 20-hour battery life... the battery was a monster carried external to the hat.
Good thing someone invented transistors, isn't it?
That's similar too, but much cooler than, the guys I used to see at a hamfest in Flagstaff back in the 80's and 90's who had a TV camera and a 1.2 Ghz yagi antenna along with a 1 or 2 watt transceiver all mounted in a backpack with a helmet cam.
They were Broadcasting a live video link of the hamfest people.
I'd rather have one of these 50's era radio hats any day of the week.