BOSTON -- The New York Yankees are facing the dispiriting prospect of being swept in a series they desperately needed to win. They find themselves eight games back in the AL East race and four games out of the second wild-card spot with 14 games left, and even their eternally optimistic manager agrees they need to win virtually all of them.

And for a vital, nationally televised game against the Boston Red Sox, they are forced to field a lineup that would seem more appropriate for a spring training game than for the late stages of a playoff race. The newly acquired Billy Butler is in the cleanup spot, backup Ronald Torreyes is starting at third base and hitting second, and Donovan Solano, called up in the morning from Triple-A, is hurriedly inserted into Sunday night's starting lineup.

"It's kind of the choices that we had," Girardi said. He had already lost starting second baseman Starlin Castro to a hamstring strain and center fielder Jacoby Ellsbury to a bone bruise on his right knee in Saturday's game before learning Sunday that Chase Headley, his starting third baseman, would be unavailable with stiffness in his lower back, a result of a chronic herniated-disk condition.

"It's really, really stiff," Girardi said of Headley's back. "It stiffened up in the game yesterday. He said that he probably would have said something during the game, but he noticed that [Rob] Refsnyder was out, Torreyes was in, and there really wasn't any place to go. After the game, he was really stiff, and he's still stiff today."

Castro was diagnosed Sunday with a Grade 1 strain of his right hamstring via MRI. While he will join the Yankees on their upcoming trip to Tampa for three games with the Rays beginning Tuesday, Girardi seemed to believe that Castro would miss at least a week.

"It really just depends on how quickly he heals," Girardi said. "Hopefully it's not too long. Hopefully it's not more than a week, but still I think that's probably pushing it. But you're hoping that he heals quick."

Still, the manager acknowledged that if he gets Castro -- who had five hits in 12 at-bats in the first three games of this series -- back at all, "It will be a bonus."

"It's not what you want," Girardi said of losing the hot-hitting Castro at this point in the season. "He's been swinging the bat extremely well, driving in runs, hitting home runs. He's been playing good at second base, but it's reality. Sometimes you play all these days in a row, things can pop up."

As for Ellsbury, Girardi characterized his status as "day to day," but with this caveat: "Being a center fielder, you have to be able to run is the bottom line."

That injury forced Girardi to use rookie Mason Williams in center field, meaning that three players in his starting lineup Sunday (including Butler and Solano) had not even been on the Yankees' roster five days ago. Still, Girardi expressed his usual faith that somehow, the Yankees would manage to sneak into the playoffs.

"I mean, we're going to continue to fight" he said. "I wouldn't do the things during the course of the game if I didn't think we were going to make the playoffs. That's the bottom line. You have to believe. It's who we are. You have to overcome challenges, adversity and everything that happens. I believe."

The reality, however, is that the Yankees would probably have to win 12 of their final 14 games -- totaling 89 wins -- to even have an outside shot at the second wild-card spot. And 11 of those 14 games come against the Red Sox, the Toronto Blue Jays and the Baltimore Orioles, against whom the Yankees are 19-27 this season.

"Well, you can just do the math, and I think we're all aware that we're going to have to win a lot more games than we lose," Girardi said. "If you go .500, you pretty much know that's not going to be enough. We know that we have to win a lot of games."