Step 1: Understanding the Concept:
Chromatin in eukaryotic cells is organized into repeating structural subunits called nucleosomes, which serve to compact genomic DNA inside the nucleus.
A nucleosome consists of a segment of double-stranded DNA wrapped around a core complex of basic proteins called histones.
Step 2: Detailed Explanation:
The nucleosome core particle is a highly conserved structure across eukaryotes.
It consists of approximately 146 base pairs of DNA wrapped in $1.67$ left-handed superhelical turns around a central histone protein core.
This central core is a histone octamer, which is assembled from four different types of core histone proteins: H2A, H2B, H3, and H4.
The octamer contains exactly two copies of each of these four core histones, structurally arranged as a central $(\text{H3-H4})_2$ tetramer flanked by two separate $(\text{H2A-H2B})$ dimers.
Histone H1 is a linker histone that binds to the entry and exit sites of the DNA on the nucleosome core to stabilize the higher-order chromatin structure, but it is not a part of the core octamer itself.
Histone H5 is a specialized linker histone found mainly in avian erythrocytes, replacing H1 in those cells.
Therefore, the core octamer is composed exclusively of two copies each of H2A, H2B, H3, and H4.
Step 3: Final Answer:
The composition of the histone octamer consists of two copies each of H2A, H2B, H3, and H4 histone proteins, which corresponds to option (B).