They probably do it by veneering all 5 sides and then mitering. The core would likely be MDF as that's very stable. You can build up multiple layers without a press by screwing and gluing them together. MDF tends to bulge when screwed so to avoid that predrill holes. You can remove the screws later before you veneer. Without a press of some kind you'd have to either hammer veneer it (risky I think because I've tried it on test pieces and had failures of the joint) or use contact cement. You'll need to limit the thickness to the thickness you can miter the corners on your saw. Dominos would be great for aligning the parts but you can also use biscuits or dowels if you find a jig that has a 45 degree angle adapter (I've seen two, the Dowelmax and another one I can't remember). Biscuits will have the most play which is why I'd tend to not risk it.