  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.
