I'm thinking again about the modern conflation of the concepts of "redemption" and "repentance" and "atonement" and how bleak of a picture it all paints when the definitions are homogenized.
What a lot of people mean when they talk about a person "redeeming themself" or (in fact and fiction) "redemption arcs," are actually repentance and atonement.
Repentance is the act of feeling regret for past behavior, of trying to change so that you can (and will) act differently going forward. Contrary to what you may have heard repeated in certain Christian circles, the literal meaning of the Ancient Greek word μετανοέω used in the Bible is not "to turn around wand walk the other way" (though that does provide a handy mental image). The elements are "change" and "to perceive with the eyes;" essentially, it means "to change your perspective." Or, more closely to how we would say it, "to have a change of the heart."
Atonement, on the other hand, is the process of actively trying to make up for wrongs and offset the bad you've done with good. Etymologically, it cleaves closely to 'uniting' and 'reconciling'—atonement is about mending what you've broken. If repentance is the inward change and the shift in personal behavior, atonement is about the active changes you make in how you interact with the world. (As a tangent, it's for this reason that I like to privately refer to what most people call characters' "redemption arcs" as "atonement arcs," simply because in the absence of a higher power that's actually much closer to what they are. But I get ahead of myself.)
Both repentance and atonement are good, noble, crucial things. And by virtue of being, respectively, an internal change and a set of actions done under your own power, they don't require any outward validation. In this, critics are wrong to say "some people are beyond redemption" in cases where they are, in fact, conflating it with repentance and atonement; it is somewhat ludicrous to make 'being sorry' and 'striving to do good' contingent on the acceptance of others. A person may be unrepentant. A person may refuse to atone. But in both these cases we are speaking of "will nots," not "cannots."
True "redemption," is contrast to both repentance and atonement, is not something that comes from within. "Redemption" speaks to an exchange, to a 'buying back'—by its nature, it appeals to a higher authority. A prisoner cannot absolve himself of his crimes, but he can (in an ideal system) be redeemed in the eyes of the law and of society by serving his just sentence. If I wrong my sister, I can atone by lowering myself and seeking her forgiveness, and in so doing find redemption. I can (if you'll allow me a small joke), redeem a coupon for a free sandwich, but I cannot simply will the coupon to be a sandwich by virtue of my being hungry. In all cases, redemption is something I must seek out and be given by an outside party. I can (and should) strive for it, but I cannot earn it—it is out of my hands.
But what of cases where there is no higher authority? What happens when I kill a man and no amount of private grief or public good deeds can bring him back or lay his accusations against me to rest? What happens if I seek forgiveness from my brother and am not given it? In a material framework, we must give up on the hope of redemption. If a man kills my friend, I may forgive him for the pain he has caused me, but I am not a high enough authority to forgive an attack that was committed against someone else. None of us are. The poor sinner remains trapped. In this, the critics (now actually speaking of redemption itself) are correct about some people being "beyond redemption."
It is only when we step outside of the secular viewpoint and present our penitence to, as C.S. Lewis says, "the party chiefly concerned, the person chiefly offended in all offences"—the one whose moral laws we have broken and who suffered personally as a direct result of our crimes—that we receive our answer. "I forgive you; now come, walk with me." In this, once again, the critics are proven wrong; no longer because they are conflating redemption with repentance or atonement, but because they themselves are not a high enough authority to forbid it. None of us are. The sinner remains freed.