Thumb sized tubers and larger for eating will store quite well in potato sacks or other paper bags. Like potatoes they need to be kept cool and not too dry – room temperature is far too warm for long term storage. Smaller tubers will shrivel quite quickly at room temperatures.
Wash tubers for storing for the next year’s crop and dry them off with a cloth. Store them in a sealed container – e.g. margarine tub or Ziploc bag. You can use vermiculite to pack them in, it should be just slightly damp with no visible water. Store the packs in the cool or the salad drawer of the fridge, even very small tubers will survive this way.