The dream of homeownership is "getting further and further from being a reachable goal for younger generations," she said. A few decades ago, people believed their kids would be homeowners, but "now we're coming to terms with the fact that perhaps most won't. And that's the new normal."

When rent is high - but still less than buying

The high cost of ownership is impacting tenants at all income levels. Even higher-income households, who in previous generations were able to save enough to buy a home, are choosing to rent today. In fact, renter households with incomes over $100,000 have been the fastest-growing segment of renters over the past decade, according to the Aspen Institute, including a growing class of millionaire renters.

Chris Salviati is staying put in his rental in San Francisco, the most expensive housing market in the country, with the median cost of rental listings topping $3,700. Few people follow the movements of the U.S. rental market as closely as Salviati, who has been an economist at Apartment List for nearly a decade. He has two degrees from Boston University.

"For anyone in San Francisco, housing costs are very top of mind," the 36-year-old told MarketWatch.