If I create a plane and try to Outset, nothing happens. If I Extrude, press Enter, and then scale up, the vertices look right, but I end up with two faces fighting to share the same space. So that's two things that don't work. Is there something that will work?
First change to edge select mode and select the edges.*
Now Extrude, but not the "normal" extrusion (which is called a Extrude Region) you want to Extrude Individual.
You can get to that by clicking the "Extrude Individual" button on the tool shelf, or by pressing AltE > Edges only.
Now, without doing anything press S. It is commonplace to cancel the translation part of the extrusion by pressing Esc or RMB , but that is unnecessary and not needed (kind of like that redundancy).
Here in a glorious full color gif is the super simple process.
* If you use the keyboard shortcut instead of the tool shelf button, the selection mode can be in Vertex or Edge mode.
First, do exactly what you did. (You can just press ES and scale it that way without completing the extrude in between.) Then press X (Delete) > Only Faces. This will delete the face you just added, which covered the whole thing, but leave the interior faces (the ones you actually want) intact.
This will leave the normals flipped; you can then use Ctrl+F >
Flip Normals to flip them back.