You can hand-solder with leaded solder onto a cardboard circuit board, but do not use a cardboard circuit board for reflow soldering. The spontaneous combustion point of paper is 451 degrees Fahrenheit, approximately the same temperature as your reflow solder point. Likewise, due to the higher temperatures required for unleaded solder, you should not use this with cardboard circuit boards either. Or, for the daring, if you are weary and careful at soldering, you can do it okay even with the higher temperatures.
What are alternatives you can use in the way of your own plastic sheets? 3D printed base is an option. If you want sheets of plastic, you can get them via 3D printing. Plastic sheets are useful, for example, if you have multiple separate circuit boards that you want to mount inside a single case.