How can supercapacitors not implode? How can supercapacitors store $5\,\mathrm{coloumbs}$ and not implode due to the enormous force between the plates ($10^{15}\,\mathrm{N}$ if the plates are $1\,\mathrm{cm}$ apart)?
 A: This is a good question. It comes down to two factors: The 'plates' have dielectric material separating them, and the effective size of the plates is large, relatively speaking.
The dielectric material has positive and negative charges that align themselves with the electric field of the electrodes. Fig 2 in this link shows a very simplified view of what goes on. The charges in the electrolyte move and align themselves with the electrodes, thus the force the electrodes experience is actually just to the local charges near the electrodes, not all the way to the opposite electrodes. Also keep in mind that unlike this figure, the electrodes are surrounded on all sides by electrolyte. So In other words, it's not a net force acting on each plate, but rather just a local force on each microscopic part of the plates.
This brings us to the next part. The electrodes are made up of a highly porous matrix of carbon and other materials, like this illustration. The effective surface area is high, on the order of 1000 $\mathrm{m}^2/\mathrm{g}$. So even though the stored charge is high, the surface charge density is low, or at least low enough for the materials to handle.
We can do some rough back-of-the-envelope calculations. Assume:

*

*1000 $\mathrm{m}^2/\mathrm{g}$ surface area.

*100 F/g capacitance.

*2.7 V breakdown voltage.

Then one obtains a charge of $q = Vc = 2.7 \times 100 = 270$ C/g. Using the naive formula for force between parallel plates:
$$
F = \frac{Q^2}{2A\epsilon_0}
$$
One obtains a pressure on the order of ~1 GPa. This is by using vacuum permittivity, which doesn't exactly apply here, but we can use it anyway for a rough estimate. 1 GPa is much below what would be required to e.g. tear apart the conductors at the molecular level.
Note: Capacitors with vacuum/air dielectric do exist, however their capacitances are very low, thus the amount of charge stored on them (and the force between the plates) is low.
