  We conducted an improved search for the simultaneous capture of two $K$-shell
electrons on the $^{124}$Xe and $^{126}$Xe nuclei with emission of two
neutrinos using 800.0 days of data from the XMASS-I detector. A novel method to
discriminate $\gamma$-ray/$X$-ray or double electron capture signals from
$\beta$-ray background using scintillation time profiles was developed for this
search. No significant signal was found when fitting the observed energy
spectra with the expected signal and background. Therefore, we set the most
stringent lower limits on the half-lives at $2.1 \times 10^{22}$ and $1.9
\times 10^{22}$ years for $^{124}$Xe and $^{126}$Xe, respectively, with 90%
confidence level. These limits improve upon previously reported values by a
factor of 4.5.
