  Double electron capture is a rare nuclear decay process in which two orbital
electrons are captured simultaneously in the same nucleus. Measurement of its
two-neutrino mode would provide a new reference for the calculation of nuclear
matrix elements whereas observation of its neutrinoless mode would demonstrate
lepton number violation. A search for two-neutrino double electron capture on
$^{124}$Xe is performed using 165.9 days of data collected with the XMASS-I
liquid xenon detector. No significant excess above background was observed and
we set a lower limit on the half-life as $4.7 \times 10^{21}$ years at 90%
confidence level. The obtained limit has ruled out parts of some theoretical
expectations. We obtain a lower limit on the $^{126}$Xe two-neutrino double
electron capture half-life of $4.3 \times 10^{21}$ years at 90% confidence
level as well.
