Mapping Center Answer:

Yes, you can do this, but you have to trace the outline of the polygon - you can't use an existing shape.  The New Polygon Text Tool will allow you to do this. I tried adding text to a shape that is similar to one in the map you attached (though I didn't format the text inside the polygon) -- you can see my results below (this took about a minute to trace):

(The blue boxes show the vertices I used to trace the shape of the lower 48 U.S. states, which I added as a layer in ArcMap.)

This part was easy. To format the text as in the map you attached, you would need to use text formatting tags -- that will take a little longer!