To inject some math into the discussion:
ThrustToKeepFlying = FlyingMass / LiftToDragRatio
PowerToKeepFlying = ThrustToKeepFlying * Velocity = Velocity * FlyingMass / LiftToDragRatio
Typically LiftToDragRatio is about 20 or so. Airplanes don't really make sense unless they are faster than other vehicles, so Velocity needs to be 100-300 m/s. (Typically, jets fly just under Mach 1, where they have the least drag/greatest power)
FlyingMass = AircraftMass + PayloadMass + EngineMass + PowersourceMass
Since we are using unobtainium to build our aircraft, it doesn't weigh anything. And we'll just say that we can fly arbitrarily large airplanes for a single passenger, so PayloadMass is essentially zero as well.
The best solar cells are about 300W/kg (http://en.wikipedia.org/wiki/Solar_panels_on_spacecraft), and the best electric engines are about 6 kW/kg. So
FlyingMass = OtherStuff + PowerToKeepFlying / 300 + PowerToKeepFlying / 6000 = OtherStuff + 0.0035 * PowerToKeepFlying
FlyingMass = OtherStuff + 0.0035 * ( 300 * FlyingMass / 20 )
FlyingMass = OtherStuff + 0.0525 * FlyingMass
OtherStuff = 0.9475 * FlyingMass
So this says that as long as your airplane and payload are under about 95% of the engine / power source mass, it is at least possible. Structures that light are not really an issue - the real issue is only flying during the day and in good weather. (And, of course, it would cost an arm and a leg!)