The title of the article is a bit misleading.
The issue isn't that he had a firearm in the home, it's that the police responded to a call from his wife that he was carrying a firearm while intoxicated and more to the point, when the police arrived he was carrying a shotgun. It was unloaded, but he was carrying it, claiming to be in the process of wiping it down to put it away.
The officers took the weapon from him and verified it was unloaded. If he gets off on appeal, the unloaded status will be why. Personally, I don't trust drunks with firearms, nor do I have much faith in their ability to clear them properly and thus be sure they are unloaded.
The law itself is worded as follows (emphasis added):
2923.15 Using weapons while intoxicated.
(A) No person, while under the influence of alcohol or any drug of abuse,
shall carry or use any firearm or dangerous ordnance.
(B) Whoever violates this section is guilty of using weapons while intoxicated, a misdemeanor of the first degree.
Effective Date: 01-01-1974.
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You'll note that the law doesn't prohibit possession, but rather carry or use of a firearm. You'll also note that this is a 45 year old law and isn't something recently passed.
The bottom line is that drunks and firearms are a bad combination and that despite the inflammatory and inaccurate headline nothing in this 45 year old law prevents possession of firearms in the home while intoxicated - you just can't carry them or use them.
To use the above example of car keys, under current DUI statute pretty much everywhere, you can in fact have your car keys in your home while intoxicated, you just can't go sit in your car with them. That's exactly analogous to this law.
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Here's the actual legal opinion describing the incident and the interpretation of the law. It's worth noting that this law also does not circumscribe the castle doctrine. In other words if you are drunk and use a firearm to defend yourself or your family while in your home, you are not prohibited from doing so by this law.
Sounds pretty reasonable to me.
http://www.supremecourt.ohio.gov/rod/docs/PDF/12/2019/2019-Ohio-916.pdf