We made stereoscopic observations of the Vela pulsar region with two of the 10 m diameter CANGAROO-III imaging atmospheric Cerenkov telescopes in 2004 January and February in a search for sub-TeV gamma rays from the pulsar and surrounding regions. We describe the observations, provide a detailed account of the calibration methods, and introduce the improved and bias-free analysis techniques employed for CANGAROO-III data. No evidence of gamma-ray emission is found from either the pulsar position or the previously reported position offset by 0.degrees 13, and the resulting upper limits are a factor of 5 less than the previously reported flux from observations with the CANGAROO-I 3.8 m telescope. Following the recent report by the H. E. S. S. group of TeV gamma-ray emission from the pulsar wind nebula, which is similar to 0.degrees 5 south of the pulsar position, we examined this region and found supporting evidence for emission extended over similar to 0.degrees 6.