I googled the longest scoreless streak for an MLB team when it was at 22 innings today. It was 48 innings set by the 1968 Cubs. That was the year of the pitcher, though, and the Cubs faced three hall of famers, Bob Gibson, Steve Carlton, and Phil Niekro over that stretch. Considering the mediocre pitchers the Cubs faced the last 3 games, it's pretty incredible how long it took them to score.
