Got any old roofing shingles? They will stop anything from growing around your shed. They're thin and after a couple of warm days will mold exactly to the surface of the soil they're laying on and since most of them have some kind of gritty surface, they hold onto mulch too. Carpeting would be another choice, but I don't like it as well becasue it tends to be so thick. If you wanted to spend money theres always that garden fabric which I don't prefer because weeds WILL grow through it, no matter how much you pay for it and if there is the slightest slope, mulch tends to slide off of it in a little rain. Cheryl

How about a thick (8-10 sheets) layer of newspaper under the mulch? Better for the soil if you ever do decide you want something to grow there, much easier to get rid of if you change your mind about wanting a barrier there, and way cheaper than landscape fabric.
The newspaper will eventually break down, but if your mulch is thick enough, you shouldn't have much problem with weeds. I have a couple areas that were done this way 2 years ago, one using hardwood mulch, the other using a neighbor's grass clippings, I have a weed popping through here and there, but they seem to be only in places that I didn't overlap the paper enough.
I've used roofing shingles on a pathway, they did keep the weeds completely blocked, and the grit on them kept the path from ever being slippery, but when I changed my mind about where I wanted to path to go, the shingles had deteriorated enough that they ripped into pieces when I pulled them up. And the soil underneath them was pretty icky/slimy.