Before even attempting this, you will need a team of 4 who can communicate and coordinate perfectly, and at least Exotic level gear.
If you're lucky, the Ogre will spawn before you get to this area so you don't have to worry about it. When you reach the area with the Doomwheels, clear the area of the rats, they'll respawn in increments of course, but clearing out the initial group will let you start the process.
Move the barrels as a team one at a time and place them NEAR the Doomwheels, don't put them INTO them as that will void this Achievement. You want to move as a team during this phase because if whoever is holding the barrel is hit even ONCE, the barrel's fuse will light up and it will explode. And the barrels do not respawn until they ALL have been placed or explode.
Once you've placed a barrel NEAR each wheel, you'll have to split up and pray that the RNG doesn't spawn any Packmasters or Gutter Runners. Once everyone is in position, place all three barrels in rapid succession then quickly get to the center area where the big exit door opens up.
The Achievement won't unlock until the end of the mission.