I wish I had a job where I didn't have to care about accuracy, context or history.
Here is a map to show some context - I didn't include corner store/bodegas or the year-round farmers markets for lack of time.
Constantino's Market - Full service grocery, first opened in 2004 with 9,500 square feet, expanded in 2010 to 14,000 square feet. Located in the Bingham Building in the Warehouse District.
Dave's Market - Full service grocery, open since 1997, 35,000 square feet. One block away is the West Side Market, open since 1912, 75,000 square feet. Located adjacent to Red Line Rapid station (one stop from downtown), and on multiple bus lines. Additional Dave's Markets located at East 40th and Woodland and East 33rd and Payne (not in the CBD but adjacent neighborhood)
Simply Food - Limited service grocery, opened in 1969, at one time 22,000 square feet, reduced to 6,000 square feet (reconfiguration of Reserve Square complex into all-apartments). Reserve Square was one of the largest residential projects in downtown Cleveland when it opened but the grocery component was ahead of its time, imho.
That said, no - downtown Cleveland doesn't have a "large" grocery store or a City Target or whatever Wal-Mart's urban format is. The development of the latter is hampered because in 2007, the 'power center' Steelyard Commons opened with both a Target and Walmart Supercenter, approximately three miles due south of downtown Cleveland. Steelyard Commons was built on the site of the former LTV Westside Works steel mill - unfortunately not a pedestrian-friendly format and not the most easily accessible by transit, although there are rumors of that changing in the future.