Explicit Countable Orthonormal Basis for $L^2(\mathbb{R})$

As in the comments, the Hermite functions will work. You can think of them as being obtained by applying Gram-Schmidt to the functions $x^n e^{- \frac{x^2}{2} }$ (up to some factors of $2$ depending on your conventions).

The point of the Fourier basis of $L^2(S^1)$ is not just that it's a countable orthonormal basis, it's also that it diagonalizes translation, or said another way, that it diagonalizes differentiation; this is why it's useful for solving differential equations, which were the original motivation for Fourier series (Fourier used them to solve the heat equation). The Hermite basis is great but it doesn't do this.

I know there's already an accepted answer, but my favorite is the brute force basis taking the set of exponentials $e^{2\pi i k(x-n)}$, each with $x$ restricted to [$n,n+1$) $\forall k,n\in \mathbb{N}$, which is countable by the same Cantor ordering of $(k,n)$ as for the rationals $k/n$. More pedantically, I guess the basis is $e^{2\pi i k(x-n)}\cdot I_{n,n+1}(x)$, where $I_{a,b}(x)$ is an indicator function for that interval.