In James Joyce's "Eveline," what is it that renders Eveline paralyzed?

This is a tough question, partly because there is no clear answer. One of the enduring confusions about James Joyce's "Eveline" is the protagonist's unwillingness to leave with her lover, Frank, and escape from the miserable life she so clearly detests. Eveline is clearly desperate to escape her backbreaking life caring for her younger siblings and living with her abusive father - "Escape!" she thinks to herself at one point. "She must escape! Frank would save her." (41) - and yet, when Frank urges her to leave with him, Joyce says Eveline "set her white face to him, passive, like a helpless animal. Her eyes gave him no sign of love or farewell or recognition" (43). What are we to make of Eveline's paradoxical behavior?

While Joyce characteristically offers no definite answers to this conundrum, there are a couple of potential solutions. For one thing, it could be that Eveline is unwilling to leave her younger siblings. It's clear that her alcoholic father is unable to take care of anyone or anything, and so Eveline functions as her younger siblings' only parental caregiver. If she were to leave, her siblings would probably suffer considerably. As such, it's possible that Eveline's paralysis at the end of the story might result from a selfless desire to protect her family from harm.

Alternatively, Eveline's paralysis could be a result of her unwillingness to rely yet again on a man for salvation. One of the most important themes of Joyce's short story is the oppression of women in Irish society. Eveline is trapped in her dour situation because she is forced to rely on a man, even a man as loathsome and useless as her father, to survive. In that case, escaping with Frank would only be more of the same; Eveline would once again be relying on a man to survive. It's possible that Eveline realizes this fact and, even though she has little to live for at home, decides not to leave with Frank in order to rebel against the sexist laws that govern her society. 

In the end, it's impossible to say for sure exactly why Eveline is paralyzed. However, the complexity of the tale ensures that plenty of possibilities exist, including those touched on in this answer.