There are infinitely many rational numbers between any two integers (or any two rationals), yet the rationals are still countable, so this reasoning doesn’t hold.
The only simple intuition for the uncountability of the reals I know of is Cantor’s diagonal argument.
I think aliexpress EMF paint has a good chance of working. All you need is some kind of conductor (usually graphite, that’s literally just pencil lead) mixed with some kind of adhesive. I’m not sure how the adhesive doesn’t stop electricity flow (maybe there’s just so much graphite it doesn’t matter), anyways you can buy graphite spray in any electronics hobby store, a paint is just that with a different consistency.
If you make a room’s walls and ceiling/floor electrically conductive (with paint or tin foil or wire mesh), you create a faraday cage. And as long as the gaps are small enough (proportional to the wavelength of the signal you want to block), nothing in those frequencies is getting in or out. A Faraday bag for your phone might be cheaper, though.
Can confirm it works fine with wine / proton
A practical application is for example in probability theory (or anywhere that deals with measures) such as this question:
If we generate a random real number from 0 to 1, what is the probability that it is rational?
Because we know that the continuum is so much larger in a sense than the set of rationals, we can answer this confidently and say the probability is zero, even though it is theoretically possible for us to get a rational number.
Statistics deals with similar scenarios quite frequently, and without it we wouldn’t have the modern scientific method.