We examined the kinematics , ionization conditions , and physical size of the absorption clouds in a z = 1.3911 damped Ly \alpha absorber ( DLA ) in the double image lensed quasar Q 0957 + 561 A , B ( separation 135 h _ { 75 } ^ { -1 } pc at the absorber redshift ) . Using HIRES/Keck spectra ( FWHM ~ { } \simeq 6.6 km s ^ { -1 } ) , we studied the Mg ii \lambda \lambda 2796 , 2803 doublet , Fe ii multiplet , and Mg i \lambda 2853 transition in absorption . Based upon the Fe ii profiles ( the Mg ii suffers from saturation ) , we defined six “ clouds ” in the system of sightline A and seven clouds in system of sightline 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 is \log N ( { \hbox { { Mg } \kern 1.0 pt { \sc i } } } ) / N ( { \hbox { { Fe } \kern 1.0 pt { \sc ii% } } } ) \simeq - 2 across the full \sim 300 km s ^ { -1 } velocity range in both systems and also spatially ( in both sightlines ) . 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 upon photoionization modeling , using the N ( { \hbox { { Mg } \kern 1.0 pt { \sc i } } } ) / N ( { \hbox { { Fe } \kern 1.0 pt { \sc ii } } } ) ratio in each cloud , we constrain the ionization parameters in the range -6.2 \leq \log U \leq - 5.1 , where the range brackets known abundance ratio and dust depletion patterns . The inferred cloud properties are densities of 2 \leq n _ { H } \leq 20 cm ^ { -3 } , and line of sight sizes of 1 \leq D \leq 25 pc . The masses of the clouds in system A are 10 \leq M / M _ { \odot } \leq 1000 and in system B are 1 \leq M / M _ { \odot } \leq 60 for spherical clouds . For planar clouds , the upper limits are 400 h _ { 75 } ^ { -2 } M _ { \odot } and 160 h _ { 75 } ^ { -2 } M _ { \odot } 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 ISM . We can not discern if the H i in this DLA cloud is in a single , cold phase or in cold+warm phases . A spherical cloud of \sim 10 pc would be limited to one of the sightlines ( 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 .