In sum, the Red Army was holding on by their figurative fingernails at points, and just about their literal ones too. The situation in November, 1942, was that the Germans occupied 9/10 of the city. This included part of the shoreline as well, in fact, having managed to force their way to the bank of the Volga in a portion of the northern and southern sectors of the city. The Red Army, however, still controlled important sections of the shoreline, split into three groups (and also interrupted by the Tsarista river, which their lines crossed), and vitally, were still able to control crossing points to keep those forces supplied.

Given their precarious position, there was nothing that could be called depth to the positions, and at their widest, the Red Army had only a few hundred yards between them and the river. Now, obviously the "thinnest" point would be at the edge of any given position where the line basically ends along the bank, but I expect you'd be more interested the thinnest over an appreciable stretch, in which case there were absolutely defensive points under 100 meters.

This map, from Roberts' book, shows the positions as they changed in the face of the major German offenses within the city. November 18th is the furthest German advance, coinciding with the onset of Operation Uranus, which would see the 6th Army encircled, and obviously turn the tide of the battle. You can see at that point how tiny the Soviet controlled bulges were. Scanning through several books, the smallest specific distance I could find mentioned was 70 meters for an actual defensive position to the river, which may be the thinnest, but really, the difference between that and one that was even twice as far wasn't that much. As the map shows, the Germans were having success in slowly pushing closer and closer, and had the Red Army not had the capability to launch Uranus, they might have been kicked out entirely soon enough, but we can only speculate.