Many popular dystopian novels like The Hunger Games, Gone, and even Fahrenheit 451 focus on life after the apocalypse. Whether a nuclear attack, widespread epidemic, or governmental collapse, a good dystopian story builds upon the aftermath of some sort of disaster. In your short story, consider the premise and background of your setting; what exactly has gone wrong? Many times in dystopian novels, the authors focus on society's reaction to the world around them. In some cases, the hardship of life brings out the best in people, but sometimes the author chooses to show how the dystopian setting brings out the worst in people (like in Brave New World or Lord of the Flies).
Since the required length of your short story is fairly short, I recommend that you choose a narrow focus that has potential for showing depth of characterization. In other words, focus on one or two characters' individual struggles with their new dystopian society; perhaps he or she fights to reunite with loved ones after being separated by nuclear attack, or maybe your character simply longs for an ice cream cone after the end of electricity. For inspiration, I recommend reading "By the Waters of Babylon" by Stephen Vincent Benet or "The Taxpayer" by Ray Bradbury.