Part of the premise of 'Saving Private Ryan' is that many of the soldiers don't think the rescue mission is a great idea. Ryan is just one soldier; and not even a particularly important soldier; and so why should so many of men risk their own lives for the chance of saving his? That sentiment is crucial to the film, and Spielberg wanted to make sure it came across on screen.
In order to achieve that, he put the full cast - and all the extras - through rigorous and physically draining military training over a period of many days. The only person who didn't have to go through the training was Damon, who stood and watched. Treating Damon as a 'special case' was a masterstroke by Spielberg, as it created a real sense of resentment that made everyone's performance that little bit more convincing.