  We observed the periodic radio transient GLEAM-X J162759.5-523504.3 (GLEAM-X
J1627) using the Chandra X-ray Observatory for about 30-ks on January 22-23,
2022, simultaneously with radio observations from MWA, MeerKAT and ATCA. Its
radio emission and 18-min periodicity led the source to be tentatively
interpreted as an extreme magnetar or a peculiar highly magnetic white dwarf.
The source was not detected in the 0.3-8 keV energy range with a 3-sigma
upper-limit on the count rate of 3x10^{-4} counts/s. No radio emission was
detected during our X-ray observations either. Furthermore, we studied the
field around GLEAM-X J1627 using archival ESO and DECam data, as well as recent
SALT observations. Many sources are present close to the position of GLEAM-X
J1627, but only two within the 2" radio position uncertainty. Depending on the
assumed spectral distribution, the upper limits converted to an X-ray
luminosity of L_{X}<6.5x10^{29} erg/s for a blackbody with temperature kT=0.3
keV, or L_{X}<9x10^{29} erg/s for a power-law with photon index Gamma = 2
(assuming a 1.3 kpc distance). Furthermore, we performed magneto-thermal
simulations for neutron stars considering crust- and core-dominated field
configurations. Based on our multi-band limits, we conclude that: i) in the
magnetar scenario, the X-ray upper limits suggest that GLEAM-X J1627 should be
older than ~1 Myr, unless it has a core-dominated magnetic field or has
experienced fast-cooling; ii) in the white dwarf scenario, we can rule out most
binary systems, a hot sub-dwarf and a hot magnetic isolated white dwarf
(T>10.000 K), while a cold isolated white dwarf is still compatible with our
limits.
