We examined the kinematics, ionization conditions, and physical size of the absorption clouds in a z = 1.3911 damped Lyα absorber (DLA) in the double-image lensed quasar Q0957+561A, B (separation 135 h pc at the absorber redshift). Using HIRES/Keck spectra (FWHM 6.6 km s^-1), we studied the Mg II λλ2796, 2803 doublet, Fe II multiplet, and Mg I λ2853 transition in absorption. Based on the Fe II profiles (the Mg II suffers from saturation), we defined six "clouds" in the system of sight line A and seven clouds in the system of sight line B. An examination of the N(v) profiles, using the apparent optical depth method, reveals no clear physical connection between the clouds in A and those in B. The observed column density ratios of all clouds are log N(Mg )/N(Fe ) -2 across the full ~300 km s^-1 velocity range in both systems and also spatially (in both sight lines). This is a remarkable uniformity not seen in Lyman limit systems. The uniformity of the cloud properties suggests that the multiple clouds are not part of a "halo." Based on photoionization modeling, using the N(Mg )/N(Fe ) ratio in each cloud, we constrain the ionization parameters in the range -6.2 ≤ log U ≤ -5.1, where the range brackets known abundance ratio and dust depletion patterns. The inferred cloud properties are densities of 2 ≤ n_H ≤ 20 cm^-3 and line-of-sight sizes of 1 ≤ D ≤ 25 pc. The masses of the clouds in system A are 10 ≤ M/M ≤ 1000 and in system B are 1 ≤ M/M ≤ 60 for spherical clouds. For planar clouds, the upper limits are 400 and 160 h M for A and B, respectively. We favor a model of the absorber in which the DLA region itself is a single cloud in this complex, which could be a parcel of gas in a galactic interstellar medium. We cannot discern whether the H I in this DLA cloud is in a single, cold phase or in cold+warm phases. A spherical cloud of ~10 pc would be limited to one of the sight lines (A) and imply a covering factor less than 0.1 for the DLA complex. We infer that the DLA cloud properties are consistent with those of lower density, cold clouds in the Galactic interstellar medium.