Ivy is a genus of 12–15 species of evergreen climbing or ground-creeping woody plants in the family Araliaceae, native to western, central and southern Europe, Macaronesia, northwestern Africa and across central-southern Asia east to Japan and Taiwan.
On level ground they remain creeping, not exceeding 5–20 cm height, but on suitable surfaces for climbing, including trees, natural rock outcrops or man-made structures such as quarry rock faces or built masonry and wooden structures, they can climb to at least 30 m above the ground.
The berries are moderately toxic. Ivy foliage contains triterpenoid saponins and falcarinol. Falcarinol is capable of inducing contact dermatitis. (source)
So can hamsters eat ivy?
No they can’t. Ivy is poisonous for hamsters to eat and should not be fed to them at any costs.