You can fold a paper to get a torus [1]. With those foldings, the distances on the torus embedded in 3D are the same as the distances on the flat paper.

It is even theoretically possible to embed the flat paper as a torus in 3D with a C^1 surface, without polyhedral edges [2,3]. However, this surface has a fractal structure.

Finally, any torus surface embedded in 3D that is at least C^2 (with a continuous second derivative) will nessecarily stretch some distances [4].

[1]: https://www.imaginary.org/hands-on/diplotori-flat-polyhedral...

[2]: https://aperiodical.com/2012/05/torus/

[3]: https://www.pnas.org/doi/full/10.1073/pnas.1118478109

[4]: https://math.stackexchange.com/questions/2291382/c2-isometri...