How to make a simple and portable photographic background board? Double as a reflector?

It is rare to have both, because the probability of using the reflector outdoors is not low, so it will be easily worn or dirty, and it will not look good when used as a background plate.

If you only shoot indoors, the background is mostly replaceable or fixed on the wall, so it is not necessary to use it with the reflector.

How big is it to make a dual-purpose reflector?

Background plates are generally seamless, otherwise it will be ugly, but for the convenience of carrying, most reflectors will be folded, and there is an uncompromising conflict between portability and practicality.