  We present Atacama Large Millimeter Array CO(3$-$2) and HCO$^+$(4$-$3)
observations covering the central $1\rlap{.}'5$$\times$$1\rlap{.}'5$ region of
the Orion Nebula Cluster (ONC). The unprecedented level of sensitivity
($\sim$0.1 mJy beam$^{-1}$) and angular resolution ($\sim$$0\rlap{.}''09
\approx 35$ AU) of these line observations enable us to search for gas-disk
detections towards the known positions of submillimeter-detected dust disks in
this region. We detect 23 disks in gas: 17 in CO(3$-$2), 17 in HCO$^+$(4$-$3),
and 11 in both lines. Depending on where the sources are located in the ONC, we
see the line detections in emission, in absorption against the warm background,
or in both emission and absorption. We spectrally resolve the gas with $0.5$ km
s$^{-1}$ channels, and find that the kinematics of most sources are consistent
with Keplerian rotation. We measure the distribution of gas-disk sizes and find
typical radii of $\sim$50-200 AU. As such, gas disks in the ONC are compact in
comparison with the gas disks seen in low-density star-forming regions. Gas
sizes are universally larger than the dust sizes. However, the gas and dust
sizes are not strongly correlated. We find a positive correlation between gas
size and distance from the massive star $\theta^1$ Ori C, indicating that disks
in the ONC are influenced by photoionization. Finally, we use the observed
kinematics of the detected gas lines to model Keplerian rotation and infer the
masses of the central pre-main-sequence stars. Our dynamically-derived stellar
masses are not consistent with the spectroscopically-derived masses, and we
discuss possible reasons for this discrepancy.
