Ever since British Airways introduced the first lie-flat seats in the mid 1990s, being able to fully stretch out on a long-haul flight has required ponying up for a first- or business-class ticket.

There are people who can sleep anywhere, any time, in any position – including upright in an economy-class airplane seat. And then there are the rest of us. Neck pillows just don’t cut it. Reclining a ...