Recent searches for the hosts of high-redshift ($z sim 4$) damped Ly$alpha$ absorbers (DLAs) have detected bright galaxies at distances of tens of kpc from the DLA. Using the FIRE-2 cosmological zoom simulations, we argue that these relatively large distances are due to a predominantly cool and neutral inner circumgalactic medium (CGM) surrounding high-redshift galaxies. The inner CGM is cool because of the short cooling time of hot gas in $lesssim10^{12}$ Msun halos, which implies that accretion and feedback energy are radiated quickly, while it is neutral due to the high volume densities and column densities at high redshift which shield cool gas from photoionization. Our analysis predicts large DLA covering factors ($gtrsim50%$) out to impact parameters $sim0.3((1 + z)/5)^{3/2} R_{rm vir}$ from the central galaxies at $z > 1$, equivalent to a physical distance of $sim 21 M_{12}^{1/3} ((1 + z)/5)^{1/2}$ kpc ($R_{rm vir}$ and $M_{12}$ are the halo virial radius and mass in units of $10^{12}$ Msun, respectively). This implies that DLA covering factors at $z sim 4$ may be comparable to unity out to a distance $sim 10$ times larger than stellar half-mass radii. A predominantly neutral inner CGM in the early universe suggests that its mass and metallicity can be directly constrained by CGM absorption surveys, without resorting to large ionization corrections as required for ionized CGM.
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