We examined the kinematics, ionization conditions, and physical size of the absorption 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 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 in A and those in B. The observed column density ratios of all 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 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 ≤ nH ≤ 20 cm-3 and line-of-sight sizes of 1 ≤ D ≤ 25 pc. The masses of the 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 in the Galactic interstellar medium.