Polyester fibers are not stretchy, but polyester can be made into stretchy fabric. If woven, polyester tends to be firm, like a bed sheet or dress pants material.
To be stretchy, polyester needs to either be knitted or blended with another fiber that is stretchy. For examples, if a person wanted to make curtains from polyester, a woven polyester would be best, but if workout leggings were desired, a knitted polyester-spandex blend would be ideal.
Polyester is a petroleum-based synthetic fiber, and has been in use since its development in the 20th century. It is easy to care for and is naturally water and fire resistant.