The structure in cells that contains DNA and protein is the nucleosome, found in the nucleus. In the nucleus of the cell, DNA is exists in the form of a long helical thread. It order to fit inside the nucleus, the DNA thread must be compressed enough to do so.
The cell accomplishes this by wrapping the DNA thread around a protein called histone. A single unit of DNA and histone octamer (called octamer because it consists of 8 proteins) is called a nucleosome.
The entire stretch of DNA in an average human cell is about 2 meters in length and would never fit inside a cell if not for histone proteins.