I have several used 55 gal. barrels with tops, that I bought at a local bee/honey place, for $18 apiece. I use them to store half-rounds of green wood, with the barrel tops loosely in place. The loose tops keeps the rain out (mostly), but lets it breathe some.

Works pretty well for me, mostly storing maple and cherry pieces. You can get a lot of spalting, and bugs, so you have to watch it carefully. Sometimes I will take the tops off for a while on a nice day, to reduce the humidity. Sometimes I have to tip a barrel over to drain it, if it has accumulated some water. Or, you could try filling a barrel with water deliberately, which some folks swear by as a storage method. I haven't tried that myself.

And if a piece of wood gets too far gone to turn, I will sometimes cut that up into spindle-sized pieces, and use vacuum-resin stabilization on it. Extreme spalting can produce some extremely colorful and interesting pieces of wood, for small projects like wine bottle stoppers, handles for kitchen items, etc.