Former president Barack Obama and his wife, Michelle, are spending their post-White House time in luxury — renting a $5.9 million, 8,200-square-foot residence on Belmont Road NW in Kalorama.

In recent months, passersby in the Kalorama neighborhood of Northwest Washington have watched a sleek contemporary building rising near the 50-year-old Washington Hilton. The Hepburn, a new luxury ...