I dislike the fact that I'm no longer the most pessimistic poster on the board. And so, just to be a little contrarian, I'll say it's not impossible for the Sox to stay in the pennant chase.
The '78 Red Sox were sort of similar to this Sox team. They got out to a good start, got to around 30 over .500 and fell apart in August to be passed by the Yankees who started the season miserably. Despite the Boston Massacre, the Red Sox managed to win their final 8 games of the season to force a one-game playoff. Here's a good summary of the AL pennant race that season.
I doubt the White Sox can come back to grab a playoff spot with exactly 1 starting pitcher. But it's not impossible. And moreover, I really don't think the Sox are a bad team. I've never seen an accomplished starting staff fall apart like this before, where no one is obviously injured. For the Sox, it really would have been better if someone had gone down and ushered McCarthy into the rotation.
In any case, I hope the Sox can somehow make a race out of it.