Unmissable achievement.
You get this achievement for solving the penultimate chapter of the game.
*** Spoiler - click to reveal ***
After going down the elevator, you are presented with the containment unit and the machine. Lots of stuff is floating around, including a cassette tape. Walk around the platform and up the stairs, where you will find a large circuit board.
In order to open the cage, you need to fill all of the spots on the board with vacuum tubes. Tubes come in two value: 5 and 10 (the 5 are yellow/green and shorter, the 10 are blue and longer...you can see the number on the base). Tubes are floating around everywhere and can be picked up, or you can get the tube fetcher device on the table to the left of the circuit board and use it to fetch them. Just point it in the general direction and pull the trigger and a bunch of them should come flying at you. Sometimes they'll disappear when you try to pull one off the device. Also...try not to point it at your panel or you can pull tubes you've already put in place off of it.
The trick with the panel is that there are 4 bars blocking the path and each has to have a specific value of vacuum tubes in its path (you can see the paths by tracing the different colored cords from the bottom to the top). There are clues about what values you need to get scattered around the room, but the values, from left to right, are 65 60 55 65. If you've having problems figuring this out, see the extra solution below.
Once the bars are all lowered, you can open the door to the containment unit. Pull your half of the locket/hexagon thing (from the very beginning of the game) out of your backpack and put it in the slot.
Once the elevator rises get onto it and enjoy the view as the achievement pops.
Note: Easy solution to a puzzle here:
*** Spoiler - click to reveal ***
If you're having problems solving the vacuum tube puzzle, there's a really easy solution. Place one with value 5 in any slot that only covers a single wire, and one with a value of 10 in any slot which covers multiple wires.