Here's a location guide for the X-01 Power Armors in Fallout 4.
Even though other's say different, I'm going to say the minimum level requirement for this is 30. While people have reported seeing X-01 pieces at slightly lower level's, 30 seems to be the safer baseline. (On the topic of level's, it does seem it's the case of the higher level you are, the better chance you have of X-01 armor to spawn.)
Most of the other requirement's I will state with each given location, but I do recommend having some decent weapons to protect yourselves and atleast Expert Hacking and Lockpicking with a master either one (This will allow you to gain access to almost all the pieces and sets).
Locations: Pieces of X-01
The following location's have a chance of spawning X-01 gear on the power armor suit's at them. (or in some case's are just there anyway).
Location: The Prydwen (Brotherhood of Steel Airship)
Requirements: Lockpicking (Master) or Hacking (Master)
Aboard the Prydwen and behind Proctor Teagen lies the X-01 Helmet, among other noteable loot such as a fatman. You will need either master hacking or lockpicking to gain access to the caged room.
Location: Energy Turbine 18-F
Up on the hill nearby there are two military ACP's and a crash plane of sorts. There's a suit near the tail of the plane.
Location: Abandoned Shack (Installation K21-B)
Requirements: Radiation Protection (Power armor for example) and the ability to kill Synths
Almost directly north of the Crater of Atom is the Abandoned Shack with a floor hatch that leads to the istallation.
Though the hatch is locked, the key is located within the shack. Once inside, travel down the to ground floor. Along the way there will be Synths who may or maynot be hostile to you (In my case they wheren't).
Note: Some people have reported of getting a full set of X-01 here. Tour practically guaranteed to get pieces of X-01 atleast (Need confirmation).
Location: National Guard Training Yard
Requirements: Expert Lockpicking and Hacking (?)
There are two possible chances of getting X-01 gear here. First is the easiest, a shipping container behind the armory has a Expert Lock on it with armor behind it.
The second is within the armory itself. But to open the armory you need to go inside the major building onsite. (Note: It maybe possible to lockpick the door)
Once inside, there's a terminal on the wall which can be hacked or accessed using the password found inside the other building.
Head's up on your way out of the armory though, upon exiting the building a Sentry Bot will be triggered to attack you.
(Note: The armor that spawns at this location is heavily influenced by your characters level.)
Location: Finch Farm
A little west of Finch's Farm, along the road, theres a military transport. Inside the green truck is a suit.
Location: Revere Satellite Array
This place is full of super mutants. After clearing them out, you can climb one of the satellites and find a suit of armor sitting in one of the satellite dishes.
Location: Goodneighbor (Big Dig Quest)
During the questline of Big Dig which starts through Bobbi No Nose in Goodneighbor, you can access a side room which has a suit of power armor in it (As well as a king Miralurk which can be a legendary so watch out for that).
Note: Make sure you break down the wall during the quest, cause if left up and you finish the quest, there is no way to break it down.
Location: Crashed Vertibird (West of Murkwater Construction Site)
Just west of Murkwater Construction Site is a crashed Vertibird with a set of power armor standing nearby. There will be a Mr. Gusty guarding the site though, but he's easy to take care of. Note: Be careful around Murkwater, the past few times I've gone there, a Queen Miralurk has been roaming about.
Locations: Full set's of X-01
The following location's have the chance of spawning a full set of X-01 armor. If you are to low of level, or unlucky, you will get no X-01 pieces and a full set of other power armor (T-45, T-60, etc)
Location: South Boston Military Checkpoint
Requirements: Hacking (Expert)
When you arrive there maybe some gunner's around, so you might wanna clear them out before going for the armor. On the outside of the main building lies a cage with a Terminal for the door which require's Expert hacking to access.
Location: 35 Court Tower
Requirements: The ability to kill two strong Robots
This location is just west of the Custom House Tower. It can be easily missed so I'll explain below.
Fast travel to Custom House Tower (Or discover it then fast travel) Then head a little west.
You should quickly come across a teal building as shown below.
Head around to the left side to get inside.
Upon entering the building, head for the elevator to the left (If you choose to explore a little beware of the laser tripwire behind the desk. It will activate the protectron and ceiling turrets.)
Once on the next floor head up to the open area where you will find 3 seal doors (Left, right and centre). When you approch the centre door, alarms will go off and an Assaultron and Sentry will emerge from the two side doors (In my case the assaultron was legendary so becareful of that).
Once defeated, hit the red buttons in the room's they robots where in to gain access to the power armor. (this was a full set of X-01 MK.3 armor.)
It is possible that sometimes armor will look like one set but be an entirely different set. See below.
Note: This guide is a current work-in-progress as more X-01 Power Armors location are not yet discovered.
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Credits: Special thanks to Rayquaza for this guide.