I don't know about any laws prohibiting possession, but under-21 handgun sales are prohibited by state law, as is merely
furnishing them except for hunting, sporting, or education. (
ORC 2923.21)
So if you lived in another state where individual sales are not restricted, bought a handgun legally, and then moved to Ohio while still under 21, you would be good. But as an Ohioan under 21, there does not appear to be a legal way to obtain a handgun of your own.
ETA: As stated, there is an exception for LEOs, as well as .mil; regular folks don't seem to have any legal way.
This double-age-standard bullshit really needs to stop, not only with guns, but with everything else too. It's preposterous that somebody can enter into legal contracts, but is still prohibited from purchasing and/or possessing certain items for another few years.