Depends on the carpet. I had some old commercial grade Berber that I just draped over clothesline strung between two posts. Compound arrows went in to the fletching on the side facing me. (60"; 30"DL).
Build a rack of 2x4s that are horizontal (think like a ladder across two sawhorses) about 2" apart and as wide as you want your target. Make the frame as tall as you want as well. Then cut your carpet to width and long enough to drape over the 2x4s, one piece per 2x4 or even 2 or 3 pieces. I would bet 5 or 6 pieces would do the trick as you would have 10 or 12 pieces of carpet hanging. And, you can put something substantial on the very back side, such as a stall mat from Tractor Supply, or maybe some rubber conveyor belting, to stop the arrows.
I have built a layered carpet target---stacking the pieces of carpet (cut 12 or 14" by 36") and then compressed to control penetration. Use the search function to find "Layered carpet target" for more info. It does the job but I think a hanging carpet target would be lighter and still get the job done.
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