There are a two main obstacles to Chris's pursuit of Ann, and they are pretty deeply ingrained in the Keller family.
First of all, Ann was the girlfriend of Larry, Chris's brother. That may feel more of an obstacle to modern audiences, however. There is a fairly lengthy tradition in previous time periods of brothers marrying the widows of their deceased brothers, to keep the family intact. Chris states that whenever he thought about who he wanted to marry, it was always Annie he thought of, but that it was not something that he would have pursued had Larry returned from the war. However, Kate still calls Ann "Larry's girl" because she refuses to acknowledge that Larry is dead.
Second, there is the issue of the past history between Chris & Ann's fathers. In the Keller family (and in the eyes of the courts), Steve Deever is the one to blame for the faulty parts being shipped. Both Joe and Kate preferred having the Deever family out of the picture - having Ann, and then George visit present challenges to that picture, which both Joe and Kate know to be a false picture.
Certainly, as the truth comes out as the play unfolds, the obstacles grow larger. How can Chris marry the girl whose father has been in jail because of his father's actions? Will he have to choose between his family and Ann? Tough questions!