With self storage the only downfall is the cost to rent the storerooms. If I can get my collection larger. I would consider digging down below my shed for about 2 metres then rebuild the shed with the lanterns in the proposed storage deck below.You'd need to have a suitable damp-proof membraine fitted if you're having a sub-floor area below ground level. You could probably do it with a membraine or two wrapped around wood panels that you then sunk into the ground - with the wood on the inside and the membraine outside. But it'd need to be done on all walls and the floor as well, and sealed so that water doesn't seep up through the joins.
That's why I only dug down about a foot and a half for my cabin at the previous house - because that then meant I had enough head room in the main area at 2.1meters high, and then had room for a loft space for lanterns without the outside height of the shed exceeding 2.8m (the limit for 'temporary buildings' which don't require planning permission) - the secret being that digging down meant the main room was sunk down by about a foot so that from outside the cabin it would seem that the floor started at -0.3m - and so the total height in effect was 3.1m
anyway, good luck with the abandoned building.