I want to project data of China. I'm not concerned with trying to preserve areal or directional relationships but preserving distances would be nice.
Does such a coordinate system exist?
I would recommend an Azimuthal Equidistant projection for what you are looking for. This type of projection:
"preserves true scale from a single specified point on the projection to all other points on the map."
The example below I set the central meridian to be near Beijeng, China for perspective.
Here are additional resources: