We study the Liouville metric associated to an approximation of a log-correlated Gaussian field with short range correlation. We show that below a parameter $γ_c >0$, the left-right length of rectangles for the Riemannian metric $e^{γφ_{0,n}} ds^2$ with various aspect ratio is concentrated with quasi-lognormal tails, that the renormalized metric is tight when $γ< \min ( γ_c, 0.4)$ and that subsequential limits are consistent with the Weyl scaling.