  We present results from {\em Chandra}/ACIS-S, {\em Spitzer}, {\em
XMM-Newton}, {\em HST}, and VLA observations of the radio hot spots, extended
environment, and nucleus of the nearby ($z$=0.0597) FR II radio galaxy 3C 33.
This is a relatively low-power FR II radio galaxy, and so we expect, {\it a
priori}, to detect a significant X-ray synchrotron component to the emission
from the hot spots. We detect X-ray emission coincident with the two knots of
polarized optical emission from the southern hot spot (SHS), as well as along
the northwest arm of this hot spot. We also detect X-ray emission from two
compact regions of the northern hot spot (NHS), as well as diffuse emission
behind the radio peak. The X-ray flux density of the region at the tip of the
southern hot spot, the most compact radio feature of the southern lobe, is
consistent with the synchrotron self-Compton (SSC) process. The X-ray flux
densities of the other three regions of the SHS and the two compact regions of
the NHS are an order of magnitude or more above the predictions from either the
SSC and inverse-Compton scattering of the CMB (IC/CMB) mechanisms, thus
strongly disfavoring these scenarios unless they are far from equipartition
($B$ $\sim$4-14 times smaller than the equipartition values). The X-ray flux
from the diffuse region behind the NHS is consistent with the IC/CMB prediction
assuming a small departure from equipartition. We conclude that the X-ray
emission is synchrotron emission from multiple populations of ultrarelativistic
electrons unless these regions are far from equipartition. There must therefore
be unresolved substructure within each knot, similar to that which is found in
Chandra observations of nearby FR I jets such as Centaurus A and 3C 66B.
