The main things to consider are proper storage materials and environmental conditions.
Since the actual supply drum is cardboard and metal I would store the items inside separately. The tissue rolls could be stored upright in an archival box that is acid-free, lignin-free, and has passed the Photographic Activity Test (P.A.T.). Laying them on their sides will cause the tissue to flatten there. You may need to add some archival board as spacers between the rolls and/or acid-free tissue to make sure the rolls don’t move around when the box is being handled. Storage like this will protect them from dust, light, and keep them from coming into contact with damaging materials like the cardboard supply drum.
Do you have any way to monitor the temperature and humidity in the storage space? There is no perfect answer about temperature and humidity, but the general recommendation is 55-75 degrees Fahrenheit and 45-55% relative humidity. That being said, materials like this should be kept as cools as possible and large fluctuations in temperature and humidity should be avoided.
Feel free to check out the Connecting to Collections Care Resources page for lots of great information on storage and environment.
Please let me know if you have any further questions.