Many micromachines use rotating shafts and other moving parts which carry a load and need fluid bearings for support. Most of them operate with air or water as the lubricating fluid. The present study analyzes air microbearing represented as an eccentric cylinder rotating in a stationary housing. The fluid mechanics and operating characteristics of microbearing are different from their larger cousins. The small length-scale may invalidate the continuum approximation in Navier-Stokes equations, and slip flow, rarefaction, compressibility and other unconventional effects may have to be taken into account. Surface effects dominate in small devices due to a high surface-volume ratio. In this study, two-dimensional eccentric-shaft journal microbearings with different eccentricities are simulated by direct simulation Monte Carlo (DSMC) code incorporated with a Volume-CAD software. The diffuse reflection model and Cercignani-Lampis-Lord (CLL) model are applied to model the molecule-surface interaction by considering the accommodation coefficients of shaft wall and housing wall separately. The distribution of mean free path in the flow field indicates that the continuum model may break down and it is necessary to carry our molecular modeling. Calculation results show that at high eccentricity and high accommodation coefficient on the housing wall (ACO) the flow may develop a recirculation region. However, the accommodation coefficient on shaft wall (ACI) does not have any effect on the occurrence of recirculation and the size of recirculation zone. There is antisymmetry of the pressure about a vertical axis, which produces a pressure force on the shaft wall. The influence of ACI to isobars is larger than that of ACO. The shear stress profile on shaft wall is big at low ACI. At the region of short clearance between the shaft wall and housing wall, it is also influenced by the surface condition of housing wall and may even change its direction at low ACO. The pressure profile is reduced in amplitude as the ACI increases, but it is enhanced a little with the increase of ACO. The ACO has great impact on the viscous force in the case of big eccentricity. With the increase of ACI, the viscous force decreases. The pressure force is high at large eccentricity. The influence of ACO to pressure force is insignificant, but the pressure force fall is enormous when ACI increases, especially for large eccentricity. The total force decreases markedly at high eccentricity when ACI increases. The ACO almost has no impact on the total force. The torque increases with ACO, but decreases with ACI. The eccentricity also has great impact on the torque, and the microbeaing may have large torque at high eccentricity. The method developed in this paper would be very useful for designing and evaluating journal microbearing.