S04E08 (Hermanos): Why is Gus allowed to live?

So far we don't know. All we do know is that Gustavo Fring immigrated from Chile to Mexico in 1986. What he did back in Chile and who he was remains a mystery. Hank investigated his past and couldn't find anything, leading him to suspect that the Pinochet regime (who ruled Chile at the time) were somehow responsible. Don Eladio's words seems to suggest that Gustavo was a person of some note back in Chile, or is at least related to someone of note. But, with only one episode left in the series, no further information is available.

No details were ever given in the show, but the implication was that Gus was part of the Pinochet government in Chile. See for confirmation this interview with the actor who played Gus:

http://seriable.com/breaking-bad-giancarlo-esposito-reveals-gus-backstory-that-might-still-come-to-light/

which includes this:

Gus was probably a guy who was a General at some point or a Lieutenant and had probably killed a lot of people — or maybe was a hero in the army.

That's about as official as it gets for this question.There's a ton of fan speculation based on the few tiny clues that made it into the show (google "Gus Fring Pinochet" to see some of it).

Another interview, this one with Vince Gilligan:

http://www.avclub.com/article/vince-gilligan-walks-us-through-season-four-of-ibr-63013

in which he says that the vagueness is intentional:

It has something to do with the Pinochet government, we think, but that’s about as close as we’ve gotten. At the end of the day, try not to nail down anything that we don’t have to.

and

I enjoy it when I’m given by the creator of a show or a movie all the elements that will keep me interested in the story, but leave a few aside so I can do a little of the work myself.

Both of those interviews are from around the end of the 4th season (when Gus was killed) and there was the possibility that we might still learn more of Gus's backstory through additional flashbacks, but that never happened so the nonspecific "associate of Pinochet" is all we have.

The series writers made the character of Gustavo Fring a Chilean as it seemed to them general Pinochet was a frightening enough character to give a Mexican drug lord, Don Eladio pause before killing him. Unfortunately, the number of Afro-Chileans (the group from where Gustavo would have to be) are a repressed minority in Chile. Being a fascist and somewhat of a racist, it is doubtful General Pinochet ever had any Afro-Chileans in in his inner circle making Fring's being Chilean rather pointless.

The most likely reason that Don Eladio didn't kill Fring was that he knew he had an established distribution network set up and it would cost money and time (and risk exposure) murdering Fring and then trying to either obtain his routes or start new ones. Since Fring hadn't stolen from him and since he could still use him, killing Gus' partner to show him who was in control and then allowing Fring to live, made it clear who was in charge of the operation, without causing any significant damage.

The don has no info on Gus' partner, so no leverage over him, which he has over Gus--knowing who Gus is. That's the reason Gus lives.