fourier analysis - Simultaneous time-frequency concentration of orthonormal sequences?

Such orthonormal bases do exist, as proved in:

Bourgain, J. A remark on the uncertainty principle for Hilbertian basis. J. Funct. Anal. 79 (1988), no. 1, 136--143 (MathSciNet link).

The theorem says that for each $rho>1/2$ there is an orthonormal basis for $L^2(mathbb{R})$ such that all of the variances of the basis elements and their Fourier transforms are less than $rho$. After the statement Bourgain remarks:

Thus Balian’s strong uncertainty principle does not hold for a nonperiodic
basis.

It is remarkable that this appears to have been discovered (rediscovered?) well after von Neumann's time. Powell proved more recently the result that Matt Hastings mentioned, namely that in such a case the sequence of means of the orthonormal basis is unbounded.

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My old answer, posted before reading Matt Hastings's comment led me to the correct question, was to the question of whether all of the variances can be finite. It was this:

Yes, because you can take an orthonormal basis in the Schwartz space by applying Gram-Schmidt to a countable $L^2$ dense subset of the Schwartz space.