If you don't dress appropriately you could get frost bite in those conditions.
Not to pick nits here, and I suppose you COULD get frostbite. But as long as you're actually running, which produces an amazing amount of body heat, it's nearly impossible to get actual frostbite at these temps. Even on exposed skin. And as someone who doesn't like running with anything more than a hat above me neck, I do it all the time here in MN. Is it comfortable at those temps? That's a personal choice. But my point would be someone shouldn't be running indoors at those temps purely out of safety concerns due to the cold.
This. If you are dressed for running and are covered up including hat, there is zero chance you'll freeze your nose or face in a 60-minute, single digits F run. If you were super underdressed, painfully cold hands (and weener) that can be painful for a week later but not frostbitten. Maybe if the wind was 100mph, but you wouldn't be able to run in that.
Cross country ski racing has a temperature cut off of racing not being allowed below -4F/-20C with worries about potential long term asthma (from breathing hard in races/hard workouts, not easy workouts) and potential frostbite. They do have higher speeds in skiing though. Ski racers sometimes use sports tape patches on the nose and cheeks at temps close to the limit. When I'm out just training on skis or foot, I don't need to cover my nose until about -15 F.
The National Weather Service puts out warnings when short exposure to bare skin is a frostbite risk. They did it for here a couple weeks ago when temps were warned to get as low as -25 F. I've never seen such a warning when it's even single digits negative F, much less single digits F.
Is it ok to run 60 minutes outside in single digits with face exposed in 5 degrees or 0 degrees F ?
No, people’s noses have been known to freeze, crack, and fall off.
Tycho Brahe, the Danish astronomer, is a famous example of someone who lost his nose. He fabricated a story that it happened during a sword fight but the truth is that he suffered the loss of his nose during a training run in severe cold. He was a 48.xx 400 m runner.