I store my cards in those 5000-card boxes. I have 4 of them and filled them up.
For my binder collection of each card, I just use Ultra Pro Deck Protector Standard Gaming Card sleeves on each card and place them in the Ultra Pro 9-card sheets. I place the sleeves card-top first to prevent dust from getting in.
I used to go all out to try the perfect sleeves for double sleeving and then fit them into 9-card sheets, but was having trouble finding the right sizes. I even purchased pricey special 9-card sheets Japan. You insert the card through a flap in the middle, so it's protected on all sides. It's great for dust prevention and even slight liquid accidents. Then it got really pricey so I dropped the double sleeving and Japanese 9-card sheets and just went with cheap Ultra Pro stuff.
I am using Snap-N-Store 3x5 Index Card Box and STEELMASTER Steel Card File Box (Coated in black with a scratch and chip-resistant powder finish) to store my cards, The full piano-hinge lid allows complete access to all contents.