They ran a bit differently than most of us do now, due to running barefoot. Barefoot runners tend to land more on the forefoot rather than the heel, which you can get away with in a squishy modern running shoe.

But as to your main question about running with the same arm and leg moving forward at once, pretty unlikely. This strange depiction is actually really common in art from many cultures and many eras. The best explanation I've read is that it stems from people posing in a running stance but not actually running tend to adopt this stance. It's easier to stand still in this posture, unlike an actual running position where you're constantly falling forward. This article has some more details.