"Shall I always be chose?" she said bitterly. "Shall I always be left behind when the Riders depart, to mind the house while they win renown, and find food and beds when they return?"
"A time may come soon," said he, "when none will return. Then there will be need of valour without renown, for none will remember the deeds that are done in the last defence of your homes. Yet the deeds will not be less valiant because they are unpraised."
And she answered: "All your words are but to say: you are a woman, and your place is in the house. But when the men have died in battle and honour, you have leave to be burned in the house, for the men will need it no more. But I am of the House of Eorl and not a serving-woman. I can ride and wield blade, and I do not fear either pain or death."
"What do you fear, lady?" he asked.
"A cage," she said. "To stay behind bars, until use and old age accept them, and all chance of doing great deeds is gone beyond recall or desire."
~
Anecdote time! When Tolkien was lecturing at Oxford, it was, to put it mildly, a male-dominated institution. By his time, women were actually allowed to take degrees (since 1920), as opposed to just sitting in on lectures for three years and then going home, but the vast majority of students were men. So some of the professors were, shall we say, a little alarmed by the presence of women in their lecture halls. There was one who refused to acknowledge the presence of female students, and if they were the only members of his audience, he'd say that as no one had turned up, there was no point in his teaching anything. Tolkien, apparently, used to get very nervous, mumble and avoid eye-contact. What I'm saying is, he can hardly be held up as a great proponent of gender equality.
I don't think you could say that he goes in for anything overtly misogynistic; rather, he prefers to avoid the Problem of Women entirely by having maybe four named female characters in the entire trilogy (I'm thinking of Arwen, Rosie, Galadriel and, naturally, Eowyn. If anyone can think of any more, I'd be pleased to amed this bit). The first two never really get any screentime to give them motivations and personalities beyond the purely superficial. Galadriel, who is pretty damn cool, gets an Actual Moment of Inner Conflict when it comes to the ring; we see what she'd do with it if it was hers (a device that's also a prominent part of the characterisation of several male characters, notably Sam and Boromir). So yes, she gets that bit of development that pushes her into being a character rather than a plot device. Then we come to Eowyn, who is so alive and impassioned, knows exactly what she wants to do and how to achieve it. OK, so essentially what she's saying is 'I want to be seen as one of the boys because I'm not like these other, weaker women.' So she's not a paragon of feminist ideals in any way. But what she does have is voice and agency. Her actions aren't dictated by the men around her. And for a woman in the fairly patriarchal society of Rohan, that's pretty damn impressive.
Aragorn gives us a neat summary of the role on women, not just in the fantasy-Anglo-Saxon culture of Rohan, but in the time that Tolkien was writing (1950s). Eowyn, far from being won over by his totally reasonable argument, eloquently and elegantly demolishes it, winning my heart forever in the process. Her slaying the Witch-King is often held up as the pinnacle of her badass-ness, but for me, this will always be her finest moment.