We present new and archival Chandra snapshot (10 ks each) observations of 15 optically identified (from the Sloan Digital Sky Survey, SDSS) Type 2 quasars at z=0.40-0.73. When combined with existing X-ray data, this work provides complete X-ray coverage for all 25 radio-quiet Type 2 quasars with logL_[OIII]>9.28 L_sun from Zakamska et al. (2003). Two targets out of 15 were not detected by Chandra and most of the remaining sources are X-ray weak, with nine having less than 10 counts in the 0.5-8keV band. Low-to-moderate quality spectral analysis was limited to three sources, whose properties are consistent with the presence of column densities in the range NH~10^22-10^23 cm^-2 in the source rest frame. If the [OIII] luminosity is a reliable proxy for the intrinsic X-ray luminosity, the current X-ray data indicate that Compton-thick quasars may hide among ~65 per cent of the SDSS Type 2 quasar population (L_{X, meas}/L_{X, [OIII]}<0.01); however, since the Type 2 quasar sample is selected on [OIII] luminosity, the estimated Compton-thick quasar fraction may be overestimated. Using archival Spitzer observations, we find that ~50 per cent of SDSS Type 2 quasars appear to be obscured by Compton-thick material based on both the L_{X, meas}/L_{X, mid-IR} (where mid-IR corresponds to rest-frame 12.3 micron) and L_{X, meas}/L_{X, [OIII]} ratios. We use this information to provide an estimate of the Compton-thick quasar number density at z=0.3-0.8, which we find is in broad agreement with the expectations from X-ray background models. Comment: 12 pages, 4 figures, 5 tables, accepted for publication in MNRAS