A lightsaber blade isn't MADE of light though, it's made of ENERGY, and emits a little bit of light in every direction while in operation. Popular misconception that the blade is made of light, but really it's more like a plasma, or the energy contained in the blade is forming a plasma in the region of the blade, and that plasma is emitting light.
I almost think of the lightsaber blade as an extremely hot plasma, with extremely energetic particles in it. The plasma would have charged particles, and the lightsaber incorporates some mechanism to keep highly-energetic/very hot ions on the outer side of the plasma (or the electrons are held on the outer surface and the ions on the interior, either way would work, but I'll keep explaining assuming ions on the outside, electrons on the inside). Since it is composed of ions and the ions are kept on the outside, the blade region would appear to carry a net electrical charge since the ions would all be on the surface, and every lightsaber would have the same ions. Since two lightsaber blades are both electrically-charged when looking from the outside (due to the ions being held on the outside), they would repel each other, and if there is a high enough ion-density, the repulsion would be strong enough that it would be impossible for a human to push the two blades together. And if the ions are hot enough, the temperature of the ions will be extremely high, and able to melt or cut anything they touch.
Of course it's science fiction, but the real explanation for why it doesn't exist is not that light keeps going for ever, it's that there's no way to generate a ton of energy in a plasma and maintain it in one place that like. Lightsaber = plasma blade that emits light, not a blade made of light.
Of course this explanation may not exactly follow official Star Wars canon, but being a scientist with experience working with plasmas, this is how I see it working.
