Old carpet generally doesn't provide enough padding. You would still need maybe a 6lb pad over it. What's underneath the carpet? Just slab or is there some type of tile? You would want to make sure that the tack strips could be installed without disturbing anything that doesn't need to be disturbed.

There is just concrete under the glued down carpet. As far as more padding, this is just going to be a closet...no need for "plushness."

Do carpet tack strips come with longer nails that would not only nail through the existing indoor-outdoor, but secure the strips as well?

have you thought about maybe just a large rug?

ripping up glued down carpet can be a pain, and I sympathize not wanting to do all that work for whats eventually to be just a closet.

you can get basically rugs in any size.

For the use you intend, I think it would be fine to put carpet over carpet. Someday a future owner will have the privilege of tearing it out and say "what the flip were they thinking???" but who cares. :-)

FWIW, one of the bedrooms in our current house had green carpet over yellow shag carpet over glue-down black pad over hardwood floors. It took a bit of elbow grease, but we got the wood floors back.

I have installed carpet over glue down commercial carpet.
What needs to be done is to cut strips of the glued carpet around the walls so the tack strip can be nailed, about 2 inch strips. It doesn't work when you try to nail the tack strip through the glued carpet. I have done many like this for various reasons. sometimes the cpt is hard to get up or the padding isn't important as in your case or sometimes people just don't want to spend money on padding.