When did technology become available to make plywood?
Plywood in the sense of cross-grained veneers glued together is actually very old, for example various sources claim the Egyptians and ancient Chinese used materials that are basically plywood although it wasn't made in quite the same way as the modern stuff is with whole logs steamed and then peeled apart by a giant knife!
Rotary-cut method, also referred to as peeling.
Plywood in the modern sense is a 19th century thing, but I believe the actual commercial production of it didn't get going until the early 20th. Sources vary somewhat on this, but it is the case that early in the 20th century plywood as we would understand it today starts to be seen used for things like cupboard backs, drawer bottoms and, quite quickly, found its way to major parts of furniture like the main field of a cupboard door (where it would have a good-quality veneer on at least the outside face, sometimes on both faces).
Much modern mass-produced furniture no longer uses plywood by the way, for the major boards it is now most common to see various grades of chipboard/particleboard, or MDF, with various laminates on the surface. These laminates are mostly melamine-coated paper but occasionally real wood veneers are used, obviously this is on higher-end stuff.