That shouldn't be too costly, would that cover the back and the side walls?
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I would find an area where you can dig a trench like, shooting lane, which is pitched back slightly towards the shooter. This will help with water not ponding in the pit. I would take the soil from the excavated area and push it towards the back of the pit and the sides, which will help build a backstop and berm up the sides. If you go 3-4 feet deep, which is not that deep you will be able to berm up another 3-4 feet with the soil from the trench. If you take a lot of the soil from the front of the trench area you can build up the back and sides even more.
Adding some supports in the trench to hold back the soil from falling back in is an option too. Railroad ties at grade, built up will add another layer of support and capture any errant shots.