Step 1: Materials
I considered harvesting elastic from old pairs of shorts, but I think they would have been too worn out to be useful.
Step 2: Size the Sheet
Lay the sheet on the table with the edges of the fabric hanging over the edges of the table. In my case the sheet was just about the right length, but you may need to cut yours to fit. This ended up being about 3 inches over each edge, but 4 would have probably been better.
The top of the fabric here will become the inside of the tablecloth at the end.
Mark the corners as shown in the pictures below. Basically, just draw lines where the table edge would have gone, had it gone all the way to the edge of the cloth.
Leaving a 1/2 inch or so for sewing, cut off the corner square. In retrospect, you could probably save the cutting until after you sew, but I get scissor-happy.
Step 3: Sew the Corners
Fold a corner so the lines you drew are now back-to-back. That lines you drew represent where the table will be, so you'll sew up the corners just outside of that line.
Trim excess fabric after sewing, and repeat for the other corners.
You could even stop after this step if you like, the tablecloth stayed pretty stable with just the boxed corners. The elastic adds a nice touch, though, and will stay on even if a strong breeze breaks through the room.
Step 4: Add the Elastic
Cut 4 pieces of elastic (I used the 1/4 inch variety). They should be about 6-10 inches long. I tried 6 inches, but I think a little longer would have been better.
Take a piece of the cut elastic, and fold it in half. Mark the crease, so you know where the middle of the elastic is. Using your sewing machine, sew the middle of the elastic onto one of our marked lines, close to the edge of the cloth.
Then, pull one end of the elastic as far as it will reach along the edge of the fabric, and sew. I used a close zig-zag stitch, which seemed to work okay.
Flip the fabric around and do the same for the other end of the elastic, along the edge of the fabric going the opposite way. Repeat for the other corners.
Step 5: Finished!
The baby changing table is now ready. Now I just have to figure out how to be a dad...