The Billionaire's Gates Revisited

In a previous puzzle, we discussed a Billionaire who had a toolshed with the configuration shown above.

Every room had a doorway to all the adjacent rooms, plus one to the outside. This billionaire wanted to be able to walk through every doorway exactly once- not missing any doorway, or going through any doorway twice.

Unfortunately for him, this proved to be impossible given the limitations of Euclidean geometry.

Our crapulent friend was not too be deterred. He has decided to have his toolshed built on the surface of a small asteroid, which engineers will shape into whatever non-Euclidean geometry is necessary.

No longer limited by a flat, or even necessarily spherical drawing board, can an asteroid/toolshed be made so that the each doorway can be passed through exactly once on a particular trip? Hint: While your solution may not leave the "surface" of the asteroid, don't concern yourself with gravity. Someone this rich can afford a space suit with magnetic boots.