I'd guess you need to store them IN something to keep out the dust, like a box or bag.

And if things get musty in your climate you will need air circulation. That rules out plastic bins with lids. Cardboard boxes should work. Plastic bags (e.g. garbage bags) might be ok if you leave the opening loose for air flow. This would be alright in an attic or something but would look untidy in a public part of the house.

So... I vote for cardboard boxes. Maybe some cute ones covered with decorative paper. Or those ones made of cloth-covered cardboard that fold flat.

Is this helpful, or am I missing something?

I enjoy arranging/ displaying artificial high-end flowers in vases only during their natural bloom time (ok, maybe shortly before, and for just a tad longer..) I've used either a drawer in an upstairs storage dresser, or one of two storage bins - one longer, meant for wrapping paper rolls, to accommodate longer stems of forsythia, delphinium, etc. Either storage system works well for me.
I place the stems loosely, wrap a few in tissue, and never have had crushed blooms. I don't display any of these loose stems for any long length of time, and so have no dust issues at all from use, but gently shake anyway before storage.

How about storing them in the same way as you would be drying real flowers -- some sort of upside down assembly, hanging from a hook, rod, or a ceiling in the closet or attic? I can see using an old bedding set to make a cone shaped covers, put the flowers inside and hang upside-down on a string from a hook . Should be dust free and not smashed since they won't be laying flat in a tight space. I would not "seal" the wide/lower opening -- so that you can look inside from below and quickly find the flowers you need

If you've ever noticed the boxes at the store used for floral displays for Memorial Day, they have an "X" cut in the box and the floral memorial is poked through the "X". I do the same thing at home.

I actually found cardboard boxes while shopping at Aldi that fit a couple lidded plastic storage boxes and cut them down to 6-8" high (tall enough for the stems to poke through). Cut a series of holes (X) in the bottom of the box (which now becomes the top of the box) far enough apart to accommodate your flowers without crowding. Place the box cut-side-up in the bottom of the plastic storage box and load with flowers. Put the lid on the storage box to keep them dust-free.

As far as musty goes, we rent condos on the beach for vacations and they all have musty issues. The last one we rented had used baking soda boxes in the linen closets. I tell you this was the least musty of any condo we have ever rented. Hope this helps.

Thanks for all your helpful replies!