I don't really find it necessary to have a storage building/area.
I have a metal trashcan beside each coop with their respective feed.
By the layers coop I have a plastic dog food storage container that holds their oyster shell. One bag lasts forever so I had to have a good way of storing it, fortunately nothing tries to break into that.
I keep whatever I'm using for "scratch" (treats, sometimes scrach, sometimes BOSS, currently deer corn - just whatever is a good deal) in a plastic 50# dog food container in the house by my back door.
I only buy bedding (pine pellets for horse stalls) on an as-needed basis so I don't store it.
I keep a garden rake leaning against the coop to rake their bedding as needed.
The "hospital crate" just sits around wherever I leave it (yeah, my yard can be messy) and the rest of the vet supplies are inside in a kitchen cabinent.
Extra food dishes (dog food bowls) generally get left lying around till I collect them into a pile inside by the back door for refilling with food scraps in the kitchen.
Brooding supplies such as feeders are kept in the plastic tote that I brood in, just don't lose the lid when there are chicks in it. hehehe
All of my brooding type lamps have dog hooks on them to hang by, so I hang them on the inside fence "wall" of my coop when not in use. Makes them handy IMO.
All the "poisons" (Sevin, fly traps, ant traps, bug spray, TomCat, Golden Marlin, etc.) are kept together in a cabinent inside, makes sense to me to keep them together.
I do keep a lot of my chicken stuff inside compared to some because well, the vet stuff is mostly stuff that works for dogs, cats, and even humans (like iodine for example) and the other pets live inside. Things like rat bait I don't have to refill/use very often so I can just remember where I leave it better if I put it all in the same place.