Is Harvard not a private school? Discussions of affirmative action/race aside, if they're a private institution, can't they do pretty much whatever they want? B

Is Harvard not a private school? Discussions of affirmative action/race aside, if they're a private institution, can't they do pretty much whatever they want? By what basis does anyone have authority/standing to sue? A private club can refuse admittance for any reason, or for no reason. Is this not the case for private schools?