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We present the analysis and interpretation of a sample of eight ASCA observations of broad absorption line quasi-stellar objects (BALQSOs). This is the first moderate-sized sample of sensitive BALQSO observations above 2 keV, and the BALQSOs in our sample are among the optically brightest known (B= 14.5–18.5). Despite the ability of 2–10 keV X-rays to penetrate large column densities, we find BALQSOs to be extremely weak sources above 2 keV, and we are only able to add two new 2–10 keV detections (0226–104 and IRAS 07598+ 6508) to those previously reported. By comparison with non-BALQSOs of similar optical continuum magnitudes, we derive the column densities needed to suppress the expected X-ray fluxes of our BALQSOs. In several cases we derive column densities≳ 5× 10 23 cm-2 for a neutral absorber with solar abundances. These are the largest X-ray column densities yet inferred for …