I'm afraid the constraints in the question as stated might be lower than intended, because it isn't difficult to fit a surface to most of 6-segment right-angled closed lines in space in a way that all angles lie in that surface.
Taking in account that the OP hasn't even stated that all right angles must turn to the same side, there is an even simpler solution: a cylinder. We just need to draw a 6-segment zigzag with right angles across a piece of paper and fold the paper in a cylinder to close the line.
If the fact the cylinder has has a hole is a problem, we can close it with a semi-sphere - not included in the photos because I didn't have at hand a ball of a suitable size.
Addition of a solution with all angles to the same side (no zigzag):
I beg your pardon for my poor drawing skills.
Please take two contiguous faces of a cube, make a circuit using all edges except for the common one, and attach an small square to each of the six vertexes, in the plane of the circuit.
Now put one band along each edge the circuit, joining the squares. Put it in a way that all turns at the vertexes keep to the same side. In the next picture I added normal vectors to the squares, twists and turns for clarification:
And since there are no complete twists, you can close the hole in the band to form a disc: