Coming from a heirloom sewing background. My take on it is.
Wash well - making sure all stains etc are out - cause they'll set permanently if they aren't out before you store them.
Rinse the nappies well so that no residue remains - of detergent or anything else.
Make sure your nappies are dry - they should smell fresh and clean - no hay smell or anything else.
Don't store in plastic - as it sweats
Try and store somewhere cool and dry - no extreme changes of temp - like a roof space or shed.
Store as you would any heirloom garment - wrapped in acid free tissue in a breathable box - I use some lavender to keep the bugs away - but not on the garments iykwim.
Check on a regular basis for creepy crawlies.
Be prepared that some of your elastic may perish if stored for extended periods.