You need: a sewing machine and thread, iron and ironing board, and 1 large rectangular piece each of oilcloth (or some other waterproof cloth, like a plastic tablecloth you aren't using anymore), fusible fleece, and something durable like canvas. You want these pieces to be all the same size. Mine is less than a metre squared and it's plenty big enough - its size was determined by the largest possible combination of the materials I had on hand.
Start by ironing your fusible fleece (I love that stuff, don't you?) to the wrong size of your canvas. If your oilcloth is as creased as mine was (it used to cover our kitchen table but has been squashed in a closet for a few years) you can flatten it out by shooting steam at it from your iron. DO NOT directly touch the iron to the oilcloth. If that still doesn't do it, cover it with a thick tea towel and iron it. Use a medium temperature and keep your iron moving very quickly. Oilcloth and irons are way too expensive to accidentally melt together.
Lay your oilcloth and canvas together, with wrong sides facing out. Starting about two thirds of the way along one side, sew a 1/2" seam around three sides and part way around the first side. you want to have a space about a foot long between where your seam starts and ends. Clip the seam allowances at the corners to a 45 degree angle.
Turn your project right sides out through the unsewn hole. Reach inside and press the seams outwards as much as you can, particularly in the corners, which will fight you all the way because you're using stiff fabrics. Iron, canvas side up, tucking the seam allowance in where hole is.
Topstitch all the way around, 1/4" from the edge. Be sure to catch the seam along where the hole is. As long as this seam allowance is turned under evenly, you'll never know the hole was there.
Ta Da! Put it under your kid and give her a bowl full of messy, messy fruit slices! Just wipe the oilcloth clean when she's done smearing her sticky little self all over it.