Of course, the size of the batteries needed will preclude carrying any passengers or cargo.
There are already electric small airplanes that take a couple people.
Airplanes are obviously the highest-hanging fruit for switching over to electric drive, but they're not impossible. On the pure electric front you're first going to see the current growth trend in small personal electric airplanes continuing and short-range business uses like crop dusting and the like grow. From there you'll move to the little short hop passenger flights between small regional airports, and then increasingly longer ranges and sizes.
There's also hybrid jet technologies being researched, wherein you greatly simplify your jet engines by removing the turbine, and instead drive the compressor with electric power; the casing becomes the stator and the compressor the rotor of an electric motor. The moving part count is greatly reduced and there's far less resistance to the exhaust gases leaving the engine, meaning more power and better fuel efficiency.
But actually, before all that, there's one electrification system that's just now hitting the market, but it's not where most people might think: the wheels. Jet engines are horribly efficient in running at low powers such as taxiing, and planes burn a lot of fuel just moving about and waiting to take off or heading to the gate - on a short flight a plane may burn 5-10% of its fuel just sitting on the ground. There are now small scale pilot projects out there that have battery-driven electric motors in the landing gear so that one doesn't have to waste all this fuel.