Is the "Pawn Box" possible in standard chess? Is it possible to create a "pawn box" as shown below (using legal moves and the help of black) in standard chess? 

The positions don't need to be exactly as shown, and there can be more pieces on the board if required.

[FEN "8/1k6/3PPP2/3P1P2/3PPP2/8/8/2K5 w - - 0 1"]


Note that this is, of course, not too difficult in horde chess as a black knight can remove the central pawn in the box.
As @BrianTowers showed, it can be done. Here’s a shorter proof game for the fun of it.

[FEN ""]

1. h4 g5 2. hxg5 f5 3. g4 Nf6 4. gxf6 d6 5. gxf5 Be6 6. f4 Bb3 7. axb3 c5 8. e4 c4 9. bxc4 d5 10. cxd5 e6 11. dxe6 Na6 12. b4 Nc5 13. bxc5 Qd5 14. c4 Bd6 15. cxd6 Rf8 16. cxd5 Rh8 17. d4