Farm radios are really neat. You are absolutely correct on how they were powered. As far as the batteries go for these.....
Basically most older radios have three different batteries: A, B & C. Later on the "C" circuit was eliminated removing the need for the "C" battery. But you still need to provide juice for the A & B.
The "A" battery provides DC power to the filament which in turn heats it up, this is a relatively low voltage but the battery itself is bigegr than the "B".
The "B" battery provides DC voltage between around 22 volts on the low side and around 90 volts on the high side. This is what is called "Plate Voltage"
If you have some eletrical/electronics knowledge, you can build your own Battery Eliminator. If you don't there are a couple of places that sell them. Probably the best one on the market right now and definitely worth the money, the ARBE III. Here's the link to the website.
http://www.arbeiii.com/