  We made stereoscopic observations of the Vela Pulsar region with two of the
10 m diameter CANGAROO-III imaging atmospheric Cherenkov telescopes in January
and February, 2004, 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 0.13 degree, and the resulting upper limits are a factor of five 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 $\sim$0.5 degree south of the
pulsar position, we examined this region and found supporting evidence for
emission extended over $\sim$0.6 degree.
