You can still purchase the original type of plywood that was used. Look for 1/8" birch faced, cabinet grade, plywood. The cabinet grade is free of flaws on one of the faces but it can be difficult to located someone who is still selling the product. Your best bet is to go to the supply sources your local cabinet making companies use. In Seattle metropolitan area I only found one source for it, Midway Plywood Home
One easier to find version of this same cabinet grade, birch plywood material from the regular types of lumber yard sources are the smaller size "door skins". But of course they are typically not wider than 36" and are not as long as a 4' x 8' sheet of plywood.
So the good news is this plywood is still being made and sold
. The bad news is finding it locally can involve (as it did for me) quite the treasure hunt to track down a supplier.