Unmechanical: Extended

On your way out (in the room after the adjustable pumps), there's a big box right before the elevator. If you happen to enter the elevator without the box, quit to Menu and re-load the game and you'll be back in the room.
You just need to drag that box onto the elevator and then go up with it. Achievement unlocked.
It's easiest to grab onto the top left corner with your tractor beam, then lean left and swing right and hold the Left Stick to the right. The momentum should be enough to start tipping the box. Repeat until it's on the elevator.
If you get enough momentum on the swing, the box will actually "over tip" and if you're quick enough you can actually just push it, from the left side, for a second rotation and save some time.

This video from "aka Scratch" demonstrates this achievement.

It must be said that if for some reason, you end up in the elevator without the box (eg. box is not fully in, door closes and tosses box out of elevator), it will still go up but the achievement will not unlock. It has just happened to me and as there is no level select, the only option is to start a new play through... Make sure the box rolls into the elevator, and stays there, that is all.

This one is near the end of Chapter 5, just after the We're In a Hurry achievement. There's a large metal box just before the elevator that you need to drag onto the elevator with you. Take a running start and grab onto the left corner as you fly to the right, which will allow you to flip it once and if you do it right with enough momentum, a second time. Do this a few times to 'roll' the box onto the elevator. The achievement unlocks midway up the ride.

Check the walkthrough video above at timestamp 0:40:04 for this one.

After Dealing with the pumps you will come to a room with a big box and an elevator. To get the Box on you need to gain momentum and grab the corner of the box, keep doing this until it is on the elevator