How do I take the companion cube with me after the level "Cold Boot" in Portal 2?

The wall past the emancipation grid (once you've gone through the door) will accept portals. Put one in there, get the cube and let the door close, and portal through.

When you place it on the button, put it as far towards the upper ledge (down and to the right in the picture) as possible. Place portals on either side of the door (GLaDOS lampshades how there isn't an emancipation field as you note), then go back and grab it. Crouch and get as close as possible; it may take some repositioning.

It awards an achievement when completed, Preservation of Mass: