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(My question is different than this one and the similar one about the free particle, so hold back on casting a close vote, please).

So, I was reading on Wikipedia, and ran into this statement in the Dirac-von Neumann axioms: The space $\mathbb{H}$ is a fixed complex Hilbert space of countable infinite dimension.

This is rather confusing to me; I use position and momentum as bases for this Hilbert space all the time, and those bases have cardinality $2^{\aleph_0}$. How is it that the space is countably infinite when we use uncountable basis sets to perform basic calculations?

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For a particle on an infinite line, position and momentum eigenstates are not a true basis for the Hilbert space; they aren't even in the Hilbert space. They are useful computational tools, but complications and contradictions can arise if you pretend that they are genuine states. To treat them properly requires the rigged Hilbert space formalism.

$L^2(\mathbb R)$ is indeed separable, however - the weighted Hermite polynomials (which are incidentally the eigenstates of the quantum harmonic oscillator) constitute a countable basis for it.

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