You sure it's not in the crab apple family? Those fruits look a lot like the apples on my Robinson Crab Apple.It is located in southern indiana.I am getting ready to cut some up and was going to try and salvage some for turning.
Crab apples are hard....cut one open and see if it's white like an apple inside if it is you have a crab apple tree.It's in my yard.I never noticed it had them berry looking things till i cut a piece off.The trunk is smooth with dime sized raised portions and the limbs are smooth bark.The berry looking things don't get any bigger than you see and they are hard.
I have a Robinson Crab Apple tree in my yard and several others in the development...the fruit is only about a half inch in diameter...I have seen other crab apple trees with larger fruit.They look more like a nut than a fruit and they get no bigger than the size of a pea.Crabapples get about 2" in size.I will try to get more pics later
The fruit should be like a choke cherry if it is hackberry...pulpy with a single seed. Should have had greenish flowers before fruiting...How big is the tree.