[daxfohl]

The canonical example is https://en.wikipedia.org/wiki/Vitali_set

You can kinda think of it as "the set of real numbers MOD the set of rationals". Kinda. And they're dorked up because the length is infinitesimal, but a countably infinite number of them add up to length 1.

For a more precise explanation see here: https://math.stackexchange.com/a/137959/287133

[daxfohl]

Also note that the construction of the above set requires the axiom of choice. And, as we all know, the axiom of choice is equivalent to the continuum hypothesis in ZFC. So that's how it all fits.