Is the story of No Country for Old Men ultimately redemptive or does Bell emerge from the story looking outdated and irrelevant?
I think there's some room for interpretation, but the general opinion, and my own, is that Bell is outdated and irrelevant at the end of the story. This is supported by several story elements.
Most significantly, if we are analyzing the characters in terms of heroic arcs and plot devices, Bell fails at his "quest", which was to save Moss, and is even further humiliated by his failure to protect Carla Jean. Throughout the story, despite his savvy and experience, he is always one step behind, following and never leading the plans laid out by Moss and Chigurh. He might have had a chance at redemption, the "justified murder" that he might have committed had he actually confronted Chigurh in the motel, but he fails at this too. In the end he more or less admits that he can't recognize the world and its morals anymore, and he questions how much of his identity is tied up in his cowardly actions during the war.
There's also much to be said for the dreams Bell relates at the end of the story, particularly because they take the place of a true climax and suggest a sort of melancholy obligation to acknowledge how useless Bell's efforts have been. Without getting into the details of the dreams (of which there are many detailed explanations that I encourage you to look up), it appears that Bell is almost anticipating death. His thoughts have turned completely from the morality and violence of his contemporary world.
The only redemptive element one might find here is that Bell is acknowledging his own inadequacy, and retiring so that neither he nor anyone else subject to his decisions will be hurt because of that inadequacy. This isn't easy for him to accept, though, nor would it be for anyone, which is at least partially why he turns to religion to explain it for him and take away some of his personal responsibility.