  We observed the young pulsar J1357--6429 with the {\it Chandra} and {\it
XMM-Newton} observatories. The pulsar spectrum fits well a combination of
absorbed power-law model ($\Gamma=1.7\pm0.6$) and blackbody model
($kT=140^{+60}_{-40}$ eV, $R\sim2$ km at the distance of 2.5 kpc). Strong
pulsations with pulsed fraction of $42%\pm5%$, apparently associated with the
thermal component, were detected in 0.3--1.1 keV. Surprisingly, pulsed fraction
at higher energies, 1.1--10 keV, appears to be smaller, $23%\pm4%$. The small
emitting area of the thermal component either corresponds to a hotter fraction
of the neutron star (NS) surface or indicates inapplicability of the simplistic
blackbody description. The X-ray images also reveal a pulsar-wind nebula (PWN)
with complex, asymmetric morphology comprised of a brighter, compact PWN
surrounded by the fainter, much more extended PWN whose spectral slopes are
$\Gamma=1.3\pm0.3$ and $\Gamma=1.7\pm0.2$, respectively. The extended PWN with
the observed flux of $\sim7.5\times10^{-13}$ erg s$^{-1}$ cm$^{-2}$ is a factor
of 10 more luminous then the compact PWN. The pulsar and its PWN are located
close to the center of the extended TeV source HESS J1356--645, which strongly
suggests that the VHE emission is powered by electrons injected by the pulsar
long ago. The X-ray to TeV flux ratio, $\sim0.1$, is similar to those of other
relic PWNe. We found no other viable candidates to power the TeV source. A
region of diffuse radio emission, offset from the pulsar toward the center of
the TeV source, could be synchrotron emission from the same relic PWN rather
than from the supernova remnant.
