  We describe the first X-ray observations of five short orbital period ($P_B <
1$ day), $\gamma$-ray emitting, binary millisecond pulsars. Four of these, PSRs
J0023+0923, J1124$-$3653, J1810+1744, and J2256$-$1024 are `black-widow'
pulsars, with degenerate companions of mass $\ll0.1 M_{\odot}$, three of which
exhibit radio eclipses. The fifth source, PSR J2215+5135, is an eclipsing
`redback' with a near Roche-lobe filling $\sim$0.2 solar mass non-degenerate
companion. Data were taken using the \textit{Chandra X-Ray Observatory} and
covered a full binary orbit for each pulsar. Two pulsars, PSRs J2215+5135 and
J2256$-$1024, show significant orbital variability while PSR J1124$-$3653 shows
marginal orbital variability. The lightcurves for these three pulsars have
X-ray flux minima coinciding with the phases of the radio eclipses. This
phenomenon is consistent with an intrabinary shock emission interpretation for
the X-rays. The other two pulsars, PSRs J0023+0923 and J1810+1744, are fainter
and do not demonstrate variability at a level we can detect in these data. All
five spectra are fit with three separate models: a power-law model, a blackbody
model, and a combined model with both power-law and blackbody components. The
preferred spectral fits yield power-law indices that range from 1.3 to 3.2 and
blackbody temperatures in the hundreds of eV. The spectrum for PSR J2215+5135
shows a significant hard X-ray component, with a large number of counts above 2
keV, which is additional evidence for the presence of intrabinary shock
emission and is similar to what has been detected in the low-mass X-ray binary
to millisecond pulsar transition object PSR J1023+0038.
