I've been thinking about this because I plan to write an article to this effect for the next issue of Spoonesaurus Magazine. So let's call this my brainstorm, my rough draft.
One of the main things I pay attention to when setting up a carving area is that I need more than the stump. Even if you go crazy and attach all sorts of clever hanging loops and pegs to your stump to store your axe and club, something that I don't actually find helpful, you will still need some horizontal surfaces nearby to put other things on. What other things? Billets of wood, spoon blanks, your saw, your club. I need about three other surfaces, all within an arm's reach of me standing at the stump.
Perhaps the best setup I ever had from this perspective was my stump under the woodshed, flanked by a table on either side and with chair and bench directly behind it. Other setups have used the spoon mule as part bench, part tool rest.
You need to be aware of where your chips will be flying when using the axe, so you don't have them making a mess somewhere that would be a problem. I'm left handed, so for me the chips end up between 9 and 12 o'clock. If you're right handed, it will be 12-3.
I found out the hard way that chunks really need to be collected out of the chips, otherwise they bind the whole pile together and make it a hundred times more difficult to shift with a manure fork or shovel. So that means a pile, or bins or barrels or milk crates. You likewise will want to have a plan for WHERE you intend to move the wood chips when they reach an unacceptable level. Maybe that's mulching something, or just piled in the woods, or used on a path.
I prefer to keep my axe buried in the stump and my saw unfolded and out, in part because I feel like they rust less that way. I always keep them undercover, and in the winter I smear them with whatever oil or oil/wax mixture I have on hand.
Likewise, it is really important that you keep a tourniquet at hand, somewhere you can reach for it immediately should you need it. Practice using it. Remember that applying a tourniquet is serious business with serious consequences, and should only be removed by someone at a hospital. Do the wrong things when you take it off and there can be problems. I keep my tourniquet in a can with some trauma shears (turns out I keep my cell phone in the pocket of the leg most likely to need the tourniquet, and I want to be able to call 911), and I keep that can right next to where I work. If this sounds far-fetched or like something you would rather not think about, I would agree. But you should plan for it and practice for it nonetheless.
I prefer axing outside but undercover. For many years I worked on our porch, and then in the driveway. Then under the woodshed, and finally under a tarp. In the winter I worked in my hoophouse, and now I finally have a shop to work in this winter. All of these spaces were small. 6x10, 10x12 ft. In the hoophouse, I found out the hard way that it is worth putting sheets of plywood down before anything else, to keep the woodchips out of the soil.
As for where I actually carve spoons, that varies much more. My tools are in a box, and I migrate to wherever it is nicest. Sometimes that's sitting cross legged under a tree or on top of the picnic table. Sometimes it's in my shop. Sometimes it's in the kitchen. Having my tools in a toolbox gives me horizontal spaces to spread my tools out, as well as keeping me from needing to pull everything out the way I would with tool roll, say. I get maximum working surface area in the smallest footprint.
My blog has evolved into a series of short essays on the nature of entrepreneurship, craftsmanship, and their overlap. If either of these topics is something you think about, you will probably like this.