It doesn't have to be at all heavy, the article mentions 6 kg. Remember that kinetic energy is proportional to the square of the velocity, so doubling rotational speed lets you cut weight by three quarters. Also, gyroscopic forces won't be a problem, you just mount the wheel horizontally.
Containing the stored energy in case of an accident likely requires some engineering thought, however. I suppose you would design the system so that it brakes the flywheel if it gets busted, converting the energy into heat just like normal braking.