Apex Legends

Armor and Helmet pickups can be found throughout the map and are divided by tiers. The golden legendary is the best variant and the one relevant to this achievement. The other tiers you can find are: purple (high), blue (mid) and white (low).
WARNING: There's no set location to find any item, so this achievement is luck dependent!
You could get lucky and unlock it quite early, or it may take you well beyond lvl 50 before you find both items in the same match. Some knowledge on how loot works will greatly improve your odds, so here's a heads up on what helped me find these items:

• Play with friends/dedicated squad if possible: Some comments lamented that their chance was robbed due to playing with randoms. There's been a solo mode for a while, but it's not permanent. Playing with a dedicated squad is the best way to avoid frustration since the party can cooperate in the search. Whoever picks it up first can drop the Armor and Helmet (highlight on the inventory screen and press ) so the rest of the team can unlock it as well (Confirmed by Tagst and Markyshizzle).

• Hot Zones: As soon as the match starts, a random location will light up a blue beacon (marked by a big blue circle on the map screen). These are guaranteed to spawn high-tier loot. They are often a highly contested location. The closer it is to the Dropship, the more players will be dropping in the Hot Zone. However i noticed that players tend not to bother with these if it's too far away from the Dropship initial location. So keep that in mind if you want to avoid confrontation right at the start of the match. The beacon (and the blue circle on the map screen) marking it's location will vanish as soon as the 1st safe zone/ring appears.
*Tip: When dropping, the Dropmaster can extend the squad's flight somewhat. After a while you'll lose altitude; to keep going, dive down a bit and go up again (think the wingsuit in Far Cry and Just Cause). Practice it a little and you may reach distances that seemed impossible otherwise.

• Supply Ships: As the match starts you can look out for one of these flying around the map. It has two icons on the map screen: it's current location and where it's gonna land (little green circle). Like Hot Zones, the closer they are to the Dropship initial location, the more contested they'll be. You don't need to wait for it to land; it's quite easy to drop on top as it's flying across the map. Another way to hop on it before it lands is to launch yourself in the air with those vertical yellow zip-lines (the ones with the big red balloon at the top). When it lands it'll spawn zip-lines for players to board it. Again caution is required when approaching as many players will be on the lookout for these. Usually more Supply Ships will spawn after a few minutes in random intervals.

• Care Packages: These are dropped occasionally throughout the match. The announcer (or a character from your squad) will notify when one's coming, so pay attention. As soon as you hear that, check your map screen: it's drop location is marked with a pulsating small blue circle. It always contains 3 mid-to-high loot and most often than not at least one will be Legendary. Beware though: they usually drop in wide open areas, so check your surroundings before running to them.

• Apex Pack's hidden bots: These are spider-like robots that when shot or punched will drop mid-to-high tier loot. They can be hard to spot as they're usually hiding in out-of-the-way places like corners, rooftops, tunnels, crevices, etc. They're the same robot you see when you open an "Apex Pack", with either purple or golden neon lights (there could be a blue one but i never seen it). If there's one nearby, you'll hear a high pitched robotic "bZzZ" every now and then. It's hard to describe, but it sticks out from the rest of the ambient sound effects. As long as you're paying attention to your surroundings, you'll notice it. Playing with headphones definitely improves your chances of finding these bots.

• Location Tiers: Most named locations on the map have tiers that'll dictate the quality of the loot you can find. As soon as you set foot on a major location, it'll show both it's name and tier on the left side of the screen (sometimes they're not tiered at all, so it'll only show it's name). If it shows "High Tier Loot" in purple, there's a high chance you'll find a legendary item lying around. Location tiers are random and rarely the same from match to match. Use that information to your advantage. If you're satisfied with your current gear/items, don't waste time scavenging low level places.

• Dead Player's loot boxes: Whenever a player die they'll drop a box containing every item in their inventory. If they had a Legendary item of any kind, the box will have a golden tint. Of course, this being a squad focused game, the presence of such a box can mean that it's late owner's teammates are nearby, specially if the player was killed recently. Always make sure that the coast is clear before looting dead players.

05/06/2020 - There is now a really cheap way to get this.
Play as Loba and make sure the map is Worlds Edge. Find a vault with both Gold Helmet and Body Armour in and then use the Black market perk to take the item out of the vault with out a vault key.

You can only take one item out per time because as soon as you take one item the market closes and causes an alarm to go off. But if you camp out/hide till the perk meter fills back up just rinse and repeat.

After months of not being able to get this, I wasn't sure if it would work and it does. Tried and tested!

Just got this today and it's probably the easiest way, but it seems to take some luck.

https://www.gamesradar.com/apex-legends-loot-vault-room-key-...

Obtaining a vault key (which you can see how in this article) and locating the vault is your best bet. There is ALWAYS a legendary helmet OR armor piece in this vault (sometimes both). Obtain a key, get to the vault, equip both the yellow colored helmet and armor piece and Achievement Unlocked!

Edit: Thank you to i am 4orty2wo for reminding me of this, but if you go to the big floating ship called Mirage's Voyage, there's ALWAYS a vault key in the middle of the ship in the "disco ball." Shoot it when it turns yellow (shortly after it's blue) and a vault key will drop. There's a vault nearby near the train yard. There's ALMOST ALWAYS a yellow helmet and armor piece in there!

To get this you need to be wearing a gold helmet and gold armor piece at the same time.
The absolute best way to do this I've found is to loot death boxes (when an enemy dies they will have a box with their loot in it). If enemies have gold/legendary items their box will have a gold shade over it.
I recommend being Gibiltar (the default defensive character) and running to the box and USE YOUR BUBBLE SHIELD ABILITY, as I have died many times looting by being shot. And hopefully you will be able to find a gold helmet and/or gold armor piece.
I didn't do this but this will also help out a lot. Use a mic/be in a party! If a teammate has a gold piece you need you can ask to borrow it and hopefully a random will be nice or id you're playing with friends they'll kindly give it to you.
cheers and good luck!

This requires you to have both legendary armor and helmet equipped at the same time. Usually, you can find these in care packages that drop from the sky, or death boxes (boxes that are left behind from dead players. In my experience, I found the legendary helmet but my teammate had a legendary armor. I asked him if I could borrow the armor for the achievement and he did as long as I gave it back in which I did. The achievement unlocked and I gave him the armor back. If you're having a hard time finding legendary equipment or even the legendary weapon attachments and your teammate(s) may have any of the items, just simply as them to borrow whatever you need so you can get the achievement and you'll give the item(s) back. You cant get any more simple than that unless you actually luck up and find what you need right off the bat.

Map: Olympus
Legend: Loba
Location: Fight Night

TLDR: Play as Loba and keep dropping at Fight Night, looting the legendary orbs in the ring and hopefully you get a legendary helmet or armor. Then, navigate to the surrounding area and use the Black Market ability to see if you can grab the other one you need.

Full Explanation:
A lot of the other guides are older and refer to other maps which aren't currently in rotation, so I figured I would write one that worked for me on Olympus. Obviously there's no guaranteed way to unlock this one, as it's based on luck, but I was able to get it with this method without too many attempts.

Loba has some handy abilities for this, her passive allows her to see Epic and Legendary items through walls and her Ultimate opens a "Black Market", allowing her and her teammates to grab any 2 items from the surrounding area without having to look for them. You can unlock her with 12000 Legend Tokens (you get 600 each time you level up), or you can buy her for 750 Apex Coins (you'd need to buy 1000 which is $9 or $10 depending on if there's a sale).

The method I used is that I would drop into the area of the map labelled as "Fight Night". The deal with this area is that if you go into the center fighting ring, you can't use your guns while you're in there (and nobody can shoot you from outside), so punching only. There are purple and golden orbs in the ring that you can punch to reveal loot. You're going to want to go for the golden ones, as they have a chance to contain a legendary helmet or armor. If you manage to get one, then head out of the ring and into the surrounding area (I like to go to the upper lookout building above the fighting ring) and use the Black Market ability. If you're lucky then you can grab both items without much of a hassle. For example, I dropped in and got a legendary helmet in the fighting ring, then when using the black market I was able to grab a legendary body armor and the achievement popped after a few seconds.

If you're only lucky enough to get 1/2, then at least you can spend the rest of the game going around to high tier loot areas and using the same black market strategy trying to find the one you're missing. This will likely take multiple attempts, sometimes you'll encounter enemies, sometimes it will be a smooth experience, etc. The good thing about this method is that you have a somewhat decent chance of at least getting one of the two items relatively quickly.