To get this set you must go to Bruma and start the mission Two Sides of the Coin by speaking to Arnora after rumors circulate about a husband stealing his wife's money. Follow all the guidelines supplied throughout the quest until you get to the part where you're supposed to kill Arnora. Once you confront her she will give you an option to show Jorundr her amulet in exchange for her life and half of his treasure, agree to do it her way. Show the amulet to Jorundr and he will tell you the location of his treasure, follow the arrow to the treasure where you will find a corrupt guard named Tyrellius Logellus. He will have a short conversation with you and then attack; once you kill him you can loot his body for a nearly full set of Chainmail Armor just missing the helmet, it also comes with a Silver Longsword. Then go to the chest and collect your loot to end the mission.
~This is a great opportunity to get this rare set of armor early in the game, Chainmail Armor usually doesn't start appearing until about level 8.