Looking good in spandex is an obvious requirement.
Batman’s in, so obviously you don’t have to have actual superpowers, but, as Ralph Dibney has pointed out, you do have to have detective skills. Lots of cool gadgets probably help.
Dying apparently doesn’t get your membership revoked, since Oliver Queen, Superman and Wonder Woman are still in.
Near as I can figure, your identity when you’re not wearing tights must be kept secret from the general public (and would somebody please explain to me how Wonder Woman and Black Canary pull this off? Neither one of them wears a mask, and they don’t have mind-bending powers that would enable them to make ordinary folk unable to notice that, “Hey, Diana looks an awful lot like that Wonder Woman chick…”
Being that the entire membership consists of people who dress up in silly outfits and go out and do things that could get them killed (especially the human ones, though the metahumans do have their vulerabilites as well), there obviously isn’t any mental health screening going on.
Having a criminal record would probably disqualify you, unless you were on the straight and narrow for a period of time sufficient to convince the senior members that you had mended your ways.
So, is there, like, an actual list of requirements for JLA membership, and of things that would disqualify you from joining the JLA?