Ohio is split into three zones, I live in zone A which mean I can buy one "Either Sex Tag" and for the past three years have offered an additional anterless tag which can only be used up to the end of November. So I can shoot two does or One buck and one doe.
The One Buck restriction is statewide...
If you start heading south though your options for taking does is increased; if you buy the tags you can shoot two does in Zone B and up to 4 does in Zone C.
If you have the opportunity to hunt an urban zone in the state you can pretty much kill does to your hearts content.