  We prove that limit-periodic Dirac operators generically have spectra of zero
Lebesgue measure and that a dense set of them have spectra of zero Hausdorff
dimension. The proof combines ideas of Avila from a Schr\"odinger setting with
a new commutation argument for generating open spectral gaps. This overcomes an
obstacle previously observed in the literature; namely, in Schr\"odinger-type
settings, translation of the spectral measure corresponds to small
$L^\infty$-perturbations of the operator data, but this is not true for Dirac
or CMV operators. The new argument is much more model-independent. To
demonstrate this, we also apply the argument to prove generic zero-measure
spectrum for CMV matrices with limit-periodic Verblunsky coefficients.
