Legendary items are the rarest of rare drops. The item's name will be orange if it is a legendary.
To obtain this you must own and equip one of these items for every inventory slot you have. Not only must each item be legendary, it also must have a minimum requirement of being level 70 to wear. This will take a lot of grinding to obtain at least 1 item for every inventory slot you have.
Helmet,
Shoulders,
Gloves,
Bracers,
Body Armor,
Belt,
Pants,
Boots,
Necklace,
2x Rings,
And both hands. This can be a 2 handed weapon, two swords, dual crossbows for the demon hunter, etc. as long as BOTH hands have a legendary item in it.
There are multiple ways of obtaining these legendary items however. You can always trade with other players. If you have a friend that also plays you can always swap items back and forth to help each other. You will most likely get a few of them while grinding out the levels to get to level 70 in the first place and the level 70 items drop much more often once you HAVE reached level 70. I have found that the fastest way to obtain these solo are to do a bunch of bounties on act 1 on normal difficulty to obtain rift fragments and then launch a rift on torment difficulty and run through this board killing everything you see. I get about 2 level 70 legendary items each run. (though not always the piece of armor i need) You can also spend blood shards in adventure mode to gamble on an item. The items you get by gambling with Kadala (in new tristam camp) are completely random and i HAVE obtained level 70 legendary items before, though not often. The last way to obtain them is to have a blacksmith build them for you, but for this you need to locate the blacksmith plan and then obtain the materials to build the item.
The moment you equip the last item in your inventory slot the achievement will unlock.
Edit: for the purposes of this achievement, it has been reported that green set items count as legendary. A level 70 green set item would work as well as normal legendary and you can mix and match and have both types of items as long as they are all level 70 items. Thanks to 'Momie' for pointing this out.