Will talking about a time when I failed hurt me in the admissions process?

Either a positive personal statement or one that tackles a mistake can work very well. Of course, you want to make sure you can explain how you learned from the failure, and that the failure did not involve what the committee might see as very risky, irresponsible behavior (such as drug or alcohol use). Beyond that, the most important consideration is: is the topic interesting to you? Will you enjoy writing about it? Will it enable you to produce a well-written statement?