I've lived in Columbus my whole life and also attended OSU for my undergrad and I'm about to ship out elsewhere, so here's the skinny on a few major points.
Car or No Car: Unless you never want to leave the campus area, you are going to want a car. We have a city bus system that is free for students, but it is far from awesome. It can get you to and from the Short North (which is an artsy/hipstery affluent urban type area with some solid dining and libations as well as some galleries) and downtown and then up into Clintonville which is where a lot of grad students live. The bus routes offer very limited hours and if you like to get a little sauced on the weekends, don't count on any buses after the bars close. Also there aren't very many grocery stores that it would be at all convenient to use without a car near campus. The city is large, but it isn't a true 'urban' city. Everything is spread out and very suburban. To sum it up, you can live and stay pretty much in the campus area and you'll be fine without a car, but it will really cut into your flexibility.
Where to live: There are three areas where most grad students live: 4th/Summit East Residential, Clintonville & Grandview. 4th/Summit area is just east of campus and is easily accessible on foot, via bike or the Campus bus system, which runs a route through the area and drops you off all over campus, it runs fairly late and is actually super convenient. This area though is a little sketchier than the other two. It's bordered to the south and the east by two pretty crime-laden areas and is generally where people have issues with break-ins and mugging. It's not rampant, but it exists. There are a lot of undergrads here and if you're looking for peace and quiet, probably not the best bet.
Clintonville is just north of campus and is a nice area that has a decent grad population as well as an older crowd. It has its own pretty decent bar/dining area on High St. and is pretty convenient to campus. This area is my recommendation. Housing is affordable on most grad stipends, there are a lot of dog-friendly parks and extremely liberal and tolerant people. It's where grown-ups live to go to school.
I can't comment that much on Grandview, it's a little pricier and a little less convenient to access campus without a car, but otherwise it's a lot like Clintonville, if not a bit more affluent. Everyone I know who has lived there has liked it, just make sure you can budget it.
Cost-of-Living: Columbus is a relatively cheap city. If you're going to live by yourself you can find a studio for as low as $350 in a not-to-ghetto area but for a decent one bedroom expect to spend between $500-600/mo. Lots of cheap eats. Columbus is also the home of every chain restaurant/store/bar ever it seems. We have some nice local establishments around the campus area, but we are dominated by the chains, so there are places to buy anything relatively cheaply from the man who is undercutting our local businesses. I survived living on campus making less than a $1000/mo for a long time.
Just a few things answering a couple of questions I've seen
Gay & Lesbian community: Columbus has a pretty large and active community in this regard. The Short North in particular caters to this crowd and you'll see a lot of business owners flying the rainbow flag, the same is true to a slightly lesser extent in Clintonville. Columbus is a Mecca in the Midwest for the Gay community and only rarely shows the lack of tolerance typical to the rural midwest.
Beer: Columbus sports a couple of Ohio-based brews, the best of which I would say is Great Lakes and they make some pretty solid beers, their IPA, Burning River is one of my favorites and it's not at all rare to find this on tap in the area. There's also Columbus Brewing Co. which makes some decent brews, but nothing to write home about. We've also got Gordon Biersch, which is decent but more prevalent in the downtown area than anywhere else. But don't worry a lot of bars carry microbrews from elsewhere, Bell's abounds.
Hope it helps, I'll check this every so often if anyone wants a local's spin. I know the place, so if you have questions that venture more than 5 miles outside of campus, I'm your guy.