I use these convenient storage boxes from Sunware.
Each box holds 200+ small tubes in the top gray part and 100 or so boxed in the layer beneath, so really convenient way of storage.
The top part also has separated compartments that makes it easier to order them by manufacturer.
And they are easily stackable on eachother.
Here's a 12AT7 box. Edit: Come to think of it. The top part is a "small part organizer" like you mentioned.
Just store them dry and clean and all is well. With output tubes / older tubes (1920's, 1930's) , it might be worth to mention sometimes "the internet" will tell you to store them vertically to prevent filament from sagging. I have no way of telling if that's actually a thing.