5 shelves. Build your sides and back out of 3/4" melamine, and use 1/2" for the shelves to save some weight on an already heavy project.
I was thinking of using the design that someone posted recently make four supports out of 1x4s on the corners, with an additional 1x4 support on the back to add stability and also act as a backstop when sliding the tubs in. That way the Melamine would only be used for the shelves, and the whole thing would be much lighter.
Using melamine I built myself a four shelf rack that holds 4x 32 quart or 8x 15 quart tubs. But I wouldn't build it any bigger than that because like has been said.. The weight. If I were to expand I'd just stack another one on top. And make it possible to dis-assemble into at least the two 4 shelf racks to move/relocate more easily.
draw a rectangle and mark the size of your sheet from your supply. Then start slicing out each piece you need. This will make it alot easier to convince them to cut it for you.
Here's the rough plans from my build might help
Best of luck my friend