thickness of the steel sheet to stop pellets

You will find if you buy the gamo targets from pyramid air that they use 1/8" steel I believe. If you shoot .25cal stuff, which is about 50FPE, then you'd want to step it up to 3/16". Also 6 yards really isn't that far and I would suggest trying to find a farther distance to shoot, but maybe that's just me, especially since the pellet hitting steel will be fairly loud.
 
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I would suggest building yourself a wooden box and fill it with rubber mulch or you could use a cardboard box for short term use(you would have to replace the box eventually).
​I think there has been a few examples of these on here that people have built. That should help to capture the pellets rather than having it fly all over on impact.

​Ted has an example of one on his tedsholdover.com webpage near the bottom of the page.
 
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You definitely want something to catch the pellets-I'm assuming the phone books will stop your pellets and the steel is insurance. 1/8" should do the trick. If this is your bedroom be conscious of possible lead dust from fragmenting pellets, it's not healthy to breathe lead dust and you spend around 8 hours a night in there so be aware! I use a cardboard box stuffed with old rags in my garage which does a good job containing the pellets, not sure how effective the phone books are at containing lead splatter.
 
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+1 on the rubber mulch to capture the pellets. A bag is cheap ($7-$10), it's quiet, safe, and easy. In my testing from 1 foot away, 7" of it stopped 20 consecutive shots (in the same spot) from a .22 springer putting out 22 FPE. For insurance, I use a piece of 1/2 plywood. Yes, the plywood may not stop the pellet by itself, but shouldn't have to - it's there just in case something actually makes it through the box (which has yet to happen to me). 1/8" steel should be more than adequate if that's the route you want to go. 
Caveats:
Rubber mulch can be in chips or shredded. You want the chips - they move more freely, thus are better at "filling the hole". 
Fill the box to the very top and shake it to settle the chips. Keep adding as they settle and make sure that it's filled right to the very top when you close the box - you want the box to be 100% full. Even when you do this, the top 2-3 inches of the box will not be as good at stopping pellets. 
 
"JohnL57"You definitely want something to catch the pellets-I'm assuming the phone books will stop your pellets and the steel is insurance. 1/8" should do the trick. If this is your bedroom be conscious of possible lead dust from fragmenting pellets, it's not healthy to breathe lead dust and you spend around 8 hours a night in there so be aware! I use a cardboard box stuffed with old rags in my garage which does a good job containing the pellets, not sure how effective the phone books are at containing lead splatter.
Hello Sir, I agree with you and also thank you for pointing out the possible harm from the lead dust that I had no idea before. Right now I put a phone book in a cardboard box which can catch the paper pieces and pellets. The steel sheet will be placed behind the box in case that the pellet blows through the phone book. 
 
"Chachoze"I would suggest building yourself a wooden box and fill it with rubber mulch or you could use a cardboard box for short term use(you would have to replace the box eventually).
​I think there has been a few examples of these on here that people have built. That should help to capture the pellets rather than having it fly all over on impact.
​Ted has an example of one on his tedsholdover.com webpage near the bottom of the page.
Hello, I was also thinking about building a target box, but I just found that there is a guy selling homemade pellet trap(20"x20"x15") for 50 bulks, and he said that he tested his 9mm on it without problem, so I may get it. Thank you!