Answer: The correct answer is-
The organism's cells contain a nucleus.
Eukarya domain possesses organisms that have a membrane bound true nucleus ( containing the genetic material of life forms that is DNA) and subcellular compartments ( also called cell organelles such as mitochondria, golgi body, endoplasmic reticulum).
Example- Plants and animals.
Thus, the physical characteristic that scientist use to classify organisms into the domain eukarya is 'nucleus'.