Wrong Sox.
If you want to talk about the two teams that have gone the longest without winning the World Series, it's the Cubs and the White Sox. The Cubs last won in 1908, and the White Sox in 1917 (a year further back than the Red Sox of 1918).
Between the Cubs and the White Sox, the city of Chicago has now gone a combined 168 seasons (not counting the strike year of 1994) without winning the World Series. This is, by far, the longest championship drought in any sport for any city. Even if you combine the Red Sox record of futility with the concurrent years of the old Boston Braves, Chicago would still more than double Boston's baseball non-championship run.
Duh! What was I thinkin'? That couldn't be true. It wouldn't be more than double. If you take the Red Sox' 83 unsuccessful tries (1919-2002, minus one for 1994) and add on the Braves' concurrent string of 33-34 years (1919-1952? '53? when was their last year in Boston?), you get about 116-17 unsuccessful tries for Boston. Chicago's 168 tries is a lot more than 116-17, but it's not more than double.
(I don't think the Braves ever won the World Series during that stretch, did they?)