Note: This pic actually shows a longer storage platform that I created out of 2 Ikea bookshelves but the idea is pretty much the same.
Carefully lay your bookcase on its side (you might want to enlist some help with this). After your plywood sheets are dry, place them on top of the bookcase so that they form a platform. Line up the seam where the two sheets meet with the middle shelf of the bookshelf’s grid. Width wise, they should overlap the bookcase by just a few inches so that there is a lip for your fingers to catch when you need to lift them up and access your stuff. Lengthwise, they should overlap the edges of the bookshelf by about 9″ on either side.