To the best of my knowledge (and I would love to hear other, perhaps more informed opinions and facts), you most certainly cannot do that in Ohio. You need to be out of town, on your property or with the permission of the owner on their property. We don't have any expanses as large as desert that offer almost useless land... if you find a large piece of property away from hustle & bustle, SOMEBODY owns it, and you can't just show up and start shooting on it. I would think you'd need to be out in the great open west to enjoy something like that anymore.
There are a few places that I can think of right off the hop that may be worth visiting.
For state-owned and ran outdoor ranges, there's one west of Lancaster at Deer Creek, just outside of Washington Courthouse. There's another one 25-30 miles due north of Columbus in Delaware. Both of these ranges are run by the Division of Wildlife. They offer 25-yard handgun ranges and 100-200 yard rifle ranges, and also have a skeet field where you can toss birds with a hand thrower or trap. You can shoot all day for a $5 pass which you purchase at any of a zillion locations all over the state... typically these places double as game-check stations. When shooting at a D.O.W. range, you must pay strict attention to the rules, specifically regarding the allowable targets. Paper only, no exceptions, and you'll get a nice $85 ticket if you bend this rule. Well, at least it was $85 when I got one back in '92!
Indoor ranges-- there's a pretty successful operation in New Albany, which is north of Lancaster. There's one just a hair north of Columbus in Powell called the Powder Room. Both of these are also smaller sized gun stores.
The biggest gun store in Columbus is certainly Vance's Shooter Supplies on Cleveland Ave. They are owned by Buckeye Outdoors which has a couple stores out east-- could be Lancaster or Buckeye Lake or somewhere out there. My favorite gun store is WELL north of town, north of Mansfield even, in a little town called Shelby. If he's so inclined, tell him to check out the Sportman's Den in Shelby. That goes for anyone interested in a helluva good place for us shooters and enthusiasts in Ohio.
Nothing, of course, will ever top having your own land out of town where you can do as you please. That's the dream, anyway. I have the next best thing-- a very close friend that's 50 miles away, and we've worked a little bit each year to build for ourselves a great little range on his 13 acres.