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Imaging of the Sunyaev-Zel'dovich (SZ) effect in galaxy clusters combined with cluster plasma x-ray diagnostics can measure the cosmic distance scale to high redshift. However, projecting the inverse-Compton scattering and x-ray emission along the cluster line-of-sight introduces systematic errors in the Hubble constant, H 0 , because the true shape of the cluster is not known. In this paper I present a study of the systematic errors in the value of H 0 , as determined by the x-ray and SZ properties of theoretical samples of triaxial isothermal "beta" model clusters, caused by projection effects and observer orientation relative to the model clusters' principal axes. I calculate three estimates for H 0 for each cluster, based on their large and small apparent angular core radii, and their arithmetic mean. I demonstrate that the estimates for H 0 for a sample of 25 clusters have limited systematic error: the 99.7% confidence intervals for the mean estimated H 0 analyzing the clusters using either their large or mean angular core radius are within ≃ 14% of the "true" (assumed) value of H 0 (and enclose it), for a triaxial beta model cluster sample possessing a distribution of apparent x-ray cluster ellipticities consistent with that of observed x-ray clusters. This limit on the systematic error in H 0 caused by cluster shape assumes that each sample beta model cluster has fixed shape; deviations from constant shape within the clusters may introduce additional uncertainty or bias into this result.