  We use more than a decade of radial velocity measurements for $\alpha$ Cen A,
B, and Proxima Centauri from HARPS, CHIRON, and UVES to identify the $M \sin i$
and orbital periods of planets that could have been detected if they existed.
At each point in a mass-period grid, we sample a simulated, Keplerian signal
with the precision and cadence of existing data and assess the probability that
the signal could have been produced by noise alone. Existing data places
detection thresholds in the classically defined habitable zones at about $M
\sin i$ of 53 M$_{\oplus}$ for $\alpha$ Cen A, 8.4 M$_{\oplus}$ for $\alpha$
Cen B, and 0.47 M$_{\oplus}$ for Proxima Centauri. Additionally, we examine the
impact of systematic errors, or "red noise" in the data. A comparison of white-
and red-noise simulations highlights quasi-periodic variability in the radial
velocities that may be caused by systematic errors, photospheric velocity
signals, or planetary signals. For example, the red-noise simulations show a
peak above white-noise simulations at the period of Proxima Centauri b. We also
carry out a spectroscopic analysis of the chemical composition of the $\alpha$
Centauri stars. The stars have super-solar metallicity with ratios of C/O and
Mg/Si that are similar to the Sun, suggesting that any small planets in the
$\alpha$ Cen system may be compositionally similar to our terrestrial planets.
Although the small projected separation of $\alpha$ Cen A and B currently
hampers extreme-precision radial velocity measurements, the angular separation
is now increasing. By 2019, $\alpha$ Cen A and B will be ideal targets for
renewed Doppler planet surveys.
