There are a number of methods that gardeners use to store and document their tubers. We have an enclosed storage room built in our basement with a temperature gauge.
You can choose to use wooden boxes, plastic tubs, cardboard boxes, and brown paper lunch bags. You can line your container with newspaper. Make sure that your containers are vented. This year, we started using small plastic tubs that have lids because they are space saving. Especially for the smaller tubers or for a variety that has a limited production. We drill holes in the sides for venting purposes.
We primarily use a mineral based medium called vermiculite. You may also choose to use peat moss or wood shavings. In our experience, we found that both these mediums tended to dry out easily so be sure to keep it lightly damp by misting with water. You should check your tubers periodically during the storage period to make sure they are not too dry or showing signs of rot. That way you can take steps to prevent any potential tuber loss. In our experience, we found that the shavings as well as the peat moss was too dry for our storage area even though we misted as needed. After experimenting over the last three years, we found that what worked best for us was vermiculite.
Most gardeners have a system set up that works best for them. We have numbers written on our storage boxes and keep records on an Excel spreadsheet so we know what variety and how many tubers are in each box along with pertinent notes. For the smaller storage containers we attach labels to the outside of the container. You can also put a plant i.d. tag in the container, or simply write on the outside of a paper bag. We use the label system since we have so many and we like to stack our containers alphabetically so we can locate a variety more easily.