If you mean fully automatic (that is, a machine gun) Ohio law
DOES permit private ownership. However, the machine gun must have
been registered with the Federal BATFE. If you mean semi-automatic,
there are no state laws that forbid it.
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More answers
All NFA firearms (Short barrel rifles, short barrel shotguns,
Machineguns, AOWs, suppressors) are legal in Ohio through the usual
federal tax stamp/background check process and filing a form.