The only way it possibly works is if they're orbiting around their common centre of gravity - e.g. each world is the other's moon - and both are orbiting around the sun. Otherwise they're either going to collide or fly past each other and get thrown off into space.
There's no particular reason to think that binary planets like that are impossible, but they would almost certainly be tidally locked to each other with a day of e.g. 28 earth days if they were at the Moon's distance and there would be constant earthquakes, volcanos, etc.
