Galaxy formation and evolution are regulated by the feedback from galactic winds. Absorption lines provide the most widely available probe of winds. However, since most data only provide information integrated along the line of sight, they do not directly constrain the radial structure of the outflows. In this paper, we present a method to directly measure the gas electron density in outflows (n ( e )), which in turn yields estimates of outflow cloud properties (e.g., density, volume filling factor, and sizes/masses). We also estimate the distance (r ( n )) from the starburst at which the observed densities are found. We focus on 22 local star-forming galaxies primarily from the COS Legacy Archive Spectroscopic SurveY (CLASSY). In half of them, we detect absorption lines from fine-structure excited transitions of Si ii (i.e., Si ii*). We determine n ( e ) from relative column densities of Si ii and Si ii*, given Si ii* originates from collisional excitation by free electrons. We find that the derived n ( e ) correlates well with the galaxy's star formation rate per unit area. From photoionization models or assuming the outflow is in pressure equilibrium with the wind fluid, we get r ( n ) similar to 1-2r (*) or similar to 5r (*), respectively, where r (*) is the starburst radius. Based on comparisons to theoretical models of multiphase outflows, nearly all of the outflows have cloud sizes large enough for the clouds to survive their interaction with the hot wind fluid. Most of these measurements are the first ever for galactic winds detected in absorption lines and, thus, will provide important constraints for future models of galactic winds.