"In bitcoin," Austin Craig repeated to the young woman behind the counter at the Lean Crust Pizza parlor in the Fort Greene neighborhood of Brooklyn, N.Y. "Can we pay in bitcoin?"

"In what?" came the reply.

Mr. Craig, 30 years old, was struggling to convince Nadia Alamgir of the existence of the virtual currency that has gained traction across the world, and whose value—after months of wild swings—on Wednesday reached records around $400 per bitcoin.

It was midway through a tricontinental odyssey taken with his wife, Beccy Bingham-Craig, 29, and a film crew documenting their travails, which started in Provo, Utah. Their mission: to live on bitcoin alone.