  We study fractal properties of support sets of the critical Liouville Quantum
Gravity (cLQG) associated with the Gaussian Free Field in planar domains.
Specifically, we completely characterize the gauge functions $\phi$ (subject to
mild monotonicity conditions) for which the cLQG admits a support set of finite
$\phi$-Hausdorff measure. As a corollary, we settle the conjecture that the
cLQG is supported on a set of vanishing Hausdorff dimension. Our proofs are
based on the fact that the cLQG describes the near-critical level sets of
Discrete Gaussian Free Field.
