Elon Musk has talked about making electric planes, so I don't think they are entirely nuts.
Thinking out loud, you might build it with minimal fuel on board at all, doing a sled launch[1] or having it run its own engines off of external power for launch. Then require beamed power throughout the flight, with the provision that the plane can still land if all ground stations fail to provide power. But the plane starts and ends with less weight than a normal plane does because it's not carrying fuel, so it just might be feasible.
Then again, you can also manufacture hydrocarbons from atmospheric CO2 given energy input.