A self healing target material that would also be easy to pull arrows out of. Sand is about the best thing I know of but it has to be angled and is also very heavy.
How angled? You could make a wooden frame that’s inclined, but too little of an incline and you’d need another material skinned over it to keep the sand in.
How about using the same kind of self healing material as art cutting boards. Microbeads of plasticizer that will mend the material. They have to be somewhat slower acting for a target because the arrows aren’t removed right away. Probably not that easy to pull the arrow out though. The material has to be strong and resilent so it doesn’t break up on impact yet still return to its’ shape when the arrow is removed. Thinking about this as I write I guess something rather soft like the gel they use on Mythbusters to simulate human flesh because it won’t shatter and it also gives you a good idea how well the arrow would have penetrated a person.
Archery ranges have what they call 3 D courses where they shoot at life size foam targets of various animals. From the size of buffalo to a small skunk. The targets are very expensive and tend to wear out with heavy traffic on the range. I know a lot of guys are on this as we speak and have been for decades. The gel does seem like the best solution if it will stand on it’s own. I also imagine it is very heavy and expensive. If A guy could solve this he would have something.
Sand?
You are shooting blunt target arrows, right? I assume outside.
Straw bales. Cheaper than hay (hay is for eating, straw for stalls/bedding and straw is cheaper) Self healing and cheap. Won’t ruin your arrows, and move the targeted one to one of the supports when it finally gives up the ghost. You can cover it with any target you choose. A simple piece of paper is fine for a target on the bales.
It’s been years, but that was my set up for a 35lb recurve at about 50 feet.
Layers of carpet laying horizontally stacked upon eachother and held tight with a ratchet belt around the outside?
I have a perfect target material in mind, but saying it would likely get me something between a warning, a suspension, a ban and a knock on the door. 
Seriously, though, straw bales are used everywhere for a reason. They seem to be the perfect combination of characteristics.
Straw bales are still the preferred target at ranges. For more portable home use shrink wrap is used frequently. The expensive animal 3 dimensional targets are where a new material would really come in handy. The vitals ( heart and lung) areas of the target are first to wear out and on better targets are replaceable.
Bolding mine. See my post #5.
What are you after here?
Sometimes we also practice with broadheads which are very hard to remove, sand works well for this. But for the most part I agree straw is hard to beat.
We use layers of foam rubber, laid out as you describe, but that’s for target blunts, I don’t think it’d work as well at pulling out for broadheads - although, there’s always just pushing it through to the other side.
Used a lot in traditional archery. Compound bows, even my ancient one from the late 90s will blow through even the tightest bound bale. I can’t see them stopping the newest bows with 300 ft/s IBO speeds.
No material will last forever, but there is a semi-selfhealing foam that my F&G club uses. It is cut into 4"x4"x3’ strips and stacked in a frame that holds the ends. It is then compressed using threaded rod that runs between the top piece and the bottom of the frame. When the center strips wear out, they are either replaced, or moved to the top/bottom and fresher strips take their place. It lasts a good long time for our range, with lots of shooters.
That sounds pretty good, I need to look into it.