The membrane-bound organelle used to store nucleic acids within a eukaryotic cell is the nucleus.

Here's why:

* Nucleic Acids: DNA and RNA, the genetic material of the cell, are considered nucleic acids.

* Nucleus: The nucleus is the control center of the eukaryotic cell and contains the cell's genetic material, primarily in the form of DNA.

* Nuclear Envelope: The nucleus is surrounded by a double membrane called the nuclear envelope, which controls the movement of molecules in and out of the nucleus.

While other organelles have roles in nucleic acid processing, the nucleus is the primary storage site for DNA and is essential for regulating gene expression and cell function.