That is the question everyone is asking after reading or viewing A Streetcar Named Desire. Tennessee Williams published his play in 1947, while Elia Kazan's movie was released in 1951. Why would Stella (Kim Hunter) stay in an abusive relationship with a man with a proven history of violence? Actually, there are a plethora of reasons for her to stay with Stanley (Marlon Brando). Any of them would have been sufficient to keep her at the Elysian Fields.
The primary reason to prevent Stella's leaving was simple logistics. Where would she go? The family home had been lost, and most of her relatives were dead. She had very few places she could run. As a married woman with a child, very few doors would have open for her. She was expected to stay with her husband, so no one would take her in.
Her safety was another reason for her to stay. As any law enforcement officer can tell you, a woman is at the greatest risk after she leaves an abusive husband or boyfriend. If she had left, Stanley would likely have gone after her. Given his possessiveness, he would likely have killed her so no one else could have her. If she stayed, he would likely not do lasting damage.
The last reason and likely to most powerful was the fact the Stella did not want to leave. When Stanley wanted to, he could be very affectionate and persuasive. Simply put, Stella did not want to leave that behind. She valued those times more than the violent episodes scared her. Stanley was her life, and she did not want to lose that.
While most modern audiences are very upset that Stella stayed with Stanley, she had very good reasons to do so. While not stated, the situation would likely cycle itself until one or the other died. That is not a happy story but a realistic one.