We present high angular resolution 1.3 mm and 850 μm dust continuum data obtained with the Submillimeter Array toward 33 Class 0 protostars in nearby clouds (distance <500 pc), which represents so far the largest survey toward protostellar binary/multiple systems. The median angular resolution in the survey is 2.''5, while the median linear resolution is approximately 600 AU. Compact dust continuum emission is observed from all sources in the sample. Twenty-one sources in the sample show signatures of binarity/multiplicity, with separations ranging from 50 AU to 5000 AU. The numbers of singles, binaries, triples, and quadruples in the sample are 12, 14, 5, and 2, respectively. The derived multiplicity frequency (MF) and companion star fraction (CSF) for Class 0 protostars are 0.64 ± 0.08 and 0.91 ± 0.05, respectively, with no correction for completeness. The derived MF and CSF in this survey are approximately two times higher than the values found in the binary surveys toward Class I young stellar objects, and approximately three (for MF) and four (for CSF) times larger than the values found among main-sequence stars, with a similar range of separations. Furthermore, the observed fraction of high-order multiple systems to binary systems in Class 0 protostars (0.50 ± 0.09) is also larger than the fractions found in Class I young stellar objects (0.31 ± 0.07) and main-sequence stars (