CLEVELAND—Joe Baur grew up in a leafy suburb a half-hour outside central Cleveland. But when it came time to settle down with his wife, the couple chose Ohio City, a neighborhood here that until recently was pockmarked by abandoned businesses and decaying homes.

Mr. Baur, a 29-year-old travel writer, and Melanie, a 28-year-old international-relations worker, love that their neighborhood is brimming with new restaurants and bars. The couple pays just over $1,000 a month for a one-bedroom loft—an amount that easily could cover...