This type of material is good for straight durable pieces of wodd, but doesn't hold up to much else. If you started to refinish it, you'd just end up with a pile of dust & glue. Screws even have a hard time staying screwed in and don't react well to torque.
About your only option would be to apply a laminate over the top of the existing material. If you used the right adhesive, you could make this work. This is probably how the original furniture was made, it's typical for particle board.
You could try sanding, filling and painting, but I think the end product would just be of very questionable quality.