I am very disappointed by the lack of discussion on pointy-eared rats, cannibalistic maggots and arborophilic swine in this thread, given the title. Nevertheless...
Regarding old age, I don't believe it is just DNA. Sure, mismatches happen and in the end if nothing else gets ya, cancer will. Besides that, telomeres shorten with time and put an effective cap on how many divisions a cell can make.
Even if telomerase was somehow active in every cell and error rate was reduced by orders of magnitude, we'd probably not live much longer than we do now. In the short term, this is because the mechanisms for maintaining a body beyond a certain age do not exist. Say, your skin sags and you get dementia, and there's just no way to reverse that. I don't suppose there's some reason such mechanisms can't exist in principle, but apparently it's much more efficient to propagate your genes through the good old two-backed-beast. I guess it makes sense, since creating another organism's worth of cells would "dilute" oxidative damage and what not, but I'm really just talking out of my backside here.
It's difficult to say how longer lifespan could evolve without knowing why the short one did, but perhaps the elves have massive, extremely redundant DNA, which has the error resistance of a RAID array? This would make them grow much slower, and probably give them ludicrous gestation periods, and create pressure for long lifespan. As to why the DNA is so paranoid... Maybe a period of intense radiation in the past or something?
By the way, how come these assholes never get prion disease? They're all like, "hey let's eat grandma dudefaces!" and then they're like "hell yeah cannibalism!". Maybe they have a terrifying immune system and absurd cell turnover rates even in the nervous system. I guess elf corpses are a ready supply of nucleotides (what would they eat besides corpses though? What has nucleotide precursors? Sea weed? Great. As if it wasn't bad enough now all they eat is *blue green algae sushi*. Whatever happened to not killing plants?), which also works with the gigabase genome thing.
Anyway, they'd probably have unusually high stem cell populations too, which would nicely solve the macroscopic tissue damage problem. And give them starfish-like regenerative powers. And... asexual reproduction? I mean, if the DNA is ginormous to begin with, maybe you could have a mechanism "bootstrapping" back to totipotent in absence of certain factors. Which would have the side effect of causing a common disease of "elf growing inside elf" whenever the factor production fails for some reason. I dunno, maybe you could put a tourniquet on an elf's arm and have another elf grow out of that?
I'm most curious as to what constantly regenerating nervous tissue would mean for psychology. Technically, it's possible for new neurons to redo the connections of an old neuron pretty closely, so memory is not necessarily lost. Thing is, after a few centuries, you would either develop horrid anterograde amnesia or retrograde amnesia. Which sounds pretty horrible (See, I dunno, Henry Molaison?). Maybe they develop a ritual system for constantly refreshing memories they want to keep, hence a need for hours-long meditation sessions every day?