Smaller cities hoping to attract startup companies are finding that hip co-working space isn’t always enough—some young professionals also want co-living spaces.

Lamp Post Properties, the real-estate unit of venture-capital firm Lamp Post Group, invested $8 million to renovate a 120-year-old, four-story historic building in downtown Chattanooga, Tenn., into apartments and is offering flexible leases to people willing to share working spaces.

Tenants...