We use high-resolution (~0farcs1) F814W Advanced Camera for Surveys (ACS) images from the Hubble Space Telescope ACS Treasury survey of the Coma cluster at z ~ 0.02 to study bars in massive disk galaxies (S0s), as well as low-mass dwarf galaxies in the core of the Coma cluster, the densest environment in the nearby universe. Our study helps to constrain the evolution of bars and disks in dense environments and provides a comparison point for studies in lower density environments and at higher redshifts. Our results are: (1) we characterize the fraction and properties of bars in a sample of 32 bright (M V <~ -18, M * > 109.5 M &sun;) S0 galaxies, which dominate the population of massive disk galaxies in the Coma core. We find that the measurement of a bar fraction among S0 galaxies must be handled with special care due to the difficulty in separating unbarred S0s from ellipticals, and the potential dilution of the bar signature by light from a relatively large, bright bulge. The results depend sensitively on the method used: the bar fraction for bright S0s in the Coma core is 50% ± 11%, 65% ± 11%, and 60% ± 11% based on three methods of bar detection, namely, strict ellipse fit criteria, relaxed ellipse fit criteria, and visual classification. (2) We compare the S0 bar fraction across different environments (the Coma core, A901/902, and Virgo) adopting the critical step of using matched samples and matched methods in order to ensure robust comparisons. We find that the bar fraction among bright S0 galaxies does not show a statistically significant variation (within the error bars of ±11%) across environments which span two orders of magnitude in galaxy number density (n ~ 300-10,000 galaxies Mpc-3) and include rich and poor clusters, such as the core of Coma, the A901/902 cluster, and Virgo. We speculate that the bar fraction among S0s is not significantly enhanced in rich clusters compared to low-density environments for two reasons. First, S0s in rich clusters are less prone to bar instabilities as they are dynamically heated by harassment and are gas poor as a result of ram pressure stripping and accelerated star formation. Second, high-speed encounters in rich clusters may be less effective than slow, strong encounters in inducing bars. (3) We also take advantage of the high resolution of the ACS (~50 pc) to analyze a sample of 333 faint (M V > -18) dwarf galaxies in the Coma core. Using visual inspection of unsharp-masked images, we find only 13 galaxies with bar and/or spiral structure. An additional eight galaxies show evidence for an inclined disk. The paucity of disk structures in Coma dwarfs suggests that either disks are not common in these galaxies or that any disks present are too hot to develop instabilities. Based on observations with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program GO10861.