Talk to one of the goblins within the goblins' shop in Jeuno when you have enough fame to trigger it... and no, I don't know how much fame that is. You'll know when he starts talking to you about being able to craft armor for you. ;) Once you do have that much fame, speaking to him lets you know which race's/gender's week it is - for that entire in-game week, that race and gender combination can go to the place that the goblin names, get a chest key from the area, and then find and unlock a chest within the same area. That gives the key item of a pattern. When you get this pattern, you must return to the goblin and speak to him. You then get a choice of the 4 pieces of race-specific armor (Magna for elves), and to get the rest, you must return to the area, lather, rinse, repeat. So yes, it's a total of four trips to the area, killing monsters to find a key, locating the chest, and then returning to Jeuno. Take an instant warp scroll each time you go into the Dungeon to save time. ;)
It's helpful to take a higher-level character along, to help kill, especially in the really dangerous areas. My own RSE was found in Gusgen Mines - I've heard that the Shakrami Maze can hold the RSE quest as well on some weeks, and I believe there may be another area as well.
Oh, and it's an in-game week, not real-life. So really if you miss one shot at the quest you can get another shot within around 3 real-life days. I believe the game week starts on Fire's Day, but don't quote me on that, please.