Have a yard? Can set something up outside. Either one of those large tents where you can paint it onto the canvas floor of the tent or cut the floor out and create it on the earth floor. Or maybe a back porch or small concrete slab? I have done operations outside on a small concrete porch by using chalk. Perhaps a gazebo you can modify, put up some walls or curtains. If you live in a cold climate and do not want to be exposed to the elements, perhaps look into a small shop building, turn the whole thing into a backyard temple. Buying one pre-fabed can be expensive but if your handy a couple of 2x4's and some OSB is cheap.
Next are garages, I have also done that using chalk. Maybe a basement? Even new homes come with a attic accessed by that pull down. Put down some MDF sheets over the insulation and make a pad in a unused corner.
Inside the home: Got an unused large closet? Perhaps you can ruin the carpet in there, or cut the carpet out of the closet and expose the slab underneath. Small closets wouldn't work but many homes have walk in's in some of the guest rooms.
Other than those options above, you can go the roll up route. Use a sheet or canvas material to paint on then roll it up when done. If you want to go a little nicer perhaps buy a cheap rug and bleach the color out of the thing, then paint or sew the circle into it.
Last idea is a bit ghetto, but put down large strips of tape onto the carpet. One after another and build a big square, then draw it on with a sharpie.
There are a lot of options if you get creative enough.
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