Basicallly just a rectangular pocket of quilted fabric. I've seen them cut so the curling iron entirely fits into the holder, so just the hot iton tube fits in and a fancy one that also included an outside pocket for the cord.
I've had one for years & it works fine when I'm traveling. It's a great thing to have.
You might be able to make one from a potholder. Look around the stores for longer ones. They do make them in odd sizes & shapes now. If you find one long enough for the curling iron's barrel, simply fold it in half lenghtwise & sew the side & bottom edges shut. The loop can be left on the open edge to hang it, if you wish.
You could also go to JoAnn Fabrics & Crafts or any other fabric store that carries heat-resistant fabrics. The JoAnn's here has a quilted fabric with that silvery stuff on it for potholders & oven mitts. It would be easy enough to make one from that. It's not a wide fabric from what I remember.
Measure the length of the curling iron's barrel & get a piece that long. Cut it so it fits around the barrel & has a bit of slack so the curling iron will slide in easily. Cut a piece of fabric for the outside of it & pin it so the heat resistant side is inside & the fabric is outside. Sew seam binding along what will be the top edge. Then fold it in half & bind the side & bottom shut. If you'd like, you can make a pocket for the rolled up cord to fit into & sew it to the outside before stitching the side & bottom shut. Make it so the pocket has plenty of room to fit in when the curling iron is not in use. It'll be baggy looking when not in use, but will look fine when it's got the iron in it.
I hope you can figure this out.
Here are a bunch of pictures
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Thak you so very much for your suggestions and pictures. I think I can do it!!