The structure that protects the genetic material of a eukaryotic cell is the nucleus. The nucleus serves as the control center of the cell, housing the cell's DNA and coordinating activities such as growth, metabolism, and reproduction. It is encased in a double membrane called the nuclear envelope, which regulates the passage of substances in and out, ensuring that the genetic material is safeguarded. The nucleus plays a crucial role in gene expression and maintaining cellular integrity, making it fundamental to life processes.