When women say "real men do X and real men do Y" it means nothing, when red pillers say that, it means nothing either. Not to mention when women say that phrase, it almost always has something to do with some betabuxx / cucked up shit, like real men pay for dates, real men date single moms, real men take care of other mens children etc.
Red pillers only say it because they are insecure, its mental masturbation for them to feel like they're better than other men.
I guess in the most traditional sense a real man is basically the opposite of women and fills masculine gender roles, but because feminists have abolished the gender roles that women always held, the ones that men are still expected to hold now, are just a burden for us.