It's easy to believe there's no grocery stores in downtown Cleveland if you don't know what you're looking for, but Constantino's Market is definitely there, at 1278 W 9th St.
On the plus side, they have a lot of local grocery, they're family owned, and the quality of the food is very good. The downside is the price- but what would you expect from a grocery store with no competition? If you want groceries without having to drive out of downtown, you probably have to hit Constantino's.
The popular consensus is that it's good for the prepared foods, but the staples just aren't worth it to buy there.