There’s already a lot of answers to this question that have been posted, so I’m going to try to focus on a few different ideas about the symbolic nature of fences in Fences.

The protagonist of Fences is Troy, a struggling pater familias. Troy’s name is an excellent, often overlooked allusion to fences. The walls of Troy, in The Iliad, are arguably the most famous fences in the history of mankind—fences that eventually came tumbling down, just as Troy’s own insecurities and struggles lead to his own breakdowns with his family. Troy the character is symbolic of the city of Troy. He holds out as long as he can against forces beyond his control, for reasons concerning his own pride, which only end up hurting him.

Gabriel is another character tied to the symbol of fences. In Biblical scholarship, Gabriel is an arc angel of heaven, a correlation August Wilson calls to mind directly with one of Gabriel’s lines near the end of the play: “You ready, Troy. I’m gonna tell St. Peter to open the gates” (2.5). The gates of heaven are fences looming over the action of the entire play—the question of the best way to live one’s life and what, at the end of one’s life, makes it meaningful.

Finally, there is the American ideal of the white picket fence. Troy’s family is part of America, yet separate from it. The white picket fence which symbolizes the ideal American family seems not to apply to them—them being non-white and their circumstances not ideal. Wilson might be making the argument that the very notion of American idealism is a lie; at least for certain, usually minority, people within the American system.

In part, the fence symbolizes Troy's futile attempt to keep death and misfortune away from himself and his family. He addresses death in the play and says, "Mr. Death. See now....I'm gonna take and build me a fence around this yard...And then I want you to stay on the other side." He believes that constructing a fence will make him immune to misfortune, but his attempts are not effective. Instead, his mistress, Alberta, dies, as does Troy, eventually. A fence cannot keep death at bay.

Building fences is also Troy's attempt to wall himself away from the outside world and from other people. He believes in his core that the world is a hostile place to an African-American man, and he thinks that his son, Cory, will not be able to play football with whites. He says, "The white man ain't gonna let him get nowhere with that football." Troy even eventually places Cory's things outside the fence when he can no longer get along with his son. Essentially, Troy walls himself off emotionally from most of the outside world, believing it is a place in which he will receive no acceptance. He also walls himself off from his family and lives in a world of emotional isolation.

The idea of the fence has a number of meanings in the play. Fences keep things in and keep things out, both literally and figuratively for the Maxson family. 

For Rose, the fence represents an opportunity to define and defend her family. 

Rose thinks the partially built fence around the house will keep her loved ones safe inside.

For Troy, the fence has a more symbolic set of meanings. 

Unable to open up to those that he loves, Troy keeps much of his emotion inside, building imaginary fences between himself and his family and friends.

Beyond symbolizing the idea of Troy's inability to communicate his emotions and his affection, fences also characterize Troy's view of the challenges he has faced in his life. 

He feels restricted by race in his professional life, as he did in his athletic career. Death also represents a significant limitation for Troy, fencing off his potential in an absolute, inevitable and definite way. 

Troy's rivalry with death can be seen as indicative of his mindset generally. Where Troy sees limits, he feels that he is being personally challenged to overcome those limits. 

When Troy played baseball, he was never content to hit the ball into the stands. His hits always had to go over the fence.

What was true for Troy as a baseball player is true for him as a husband and father as well. When Troy feels powerless or limited, he acts out. We see this in his assertions of power over Cory (demonstrating that he will not be limited in his authority as a parent) and we see it as well in his adultery.

Every limit he perceives, he seeks to defeat somehow, including the limit of death.

What is the symbolic role of fences in this play?

Troy takes pride in keeping his family together and providing for them by collecting garbage. Troy's rebellion and frustration set the tone for the play as he struggles for fairness in a society which seems to offer none—and it is race that interferes with fairness, race which is the ultimate fence in society preventing people of color from attaining The American Dream. In his struggle to deal with the fence of race, he builds fences between himself and family. I don’t think in this play that “fences do good neighbors make,” for they seem to be an impediment that must be negotiated or torn down.

What is the symbolic role of fences in this play?

Rose believes the fences are symbolic of the safety they provide inside their house. Alberto wants the fences to hold death at bay, and Troy thinks the fences keep intruders out. Yet, for all that the fences symbolically keep out, they also can be seen to keep people confined, such as the imaginary fence that Troy's father uses to control him, or the prison walls that keep him incarcerated.

What is the symbolic role of fences in this play?

Just with the Robert Frost poem "Good Fences Make Good Neighbors," this play illustrates how people "fence" themselves in to certain roles in life.  Sometimes these roles are within the family, sometimes within society as a whole.  Basically, the symbolic role represents the boxes we paint for ourselves and within which we choose to live.