Step 1: Understanding the Concept:
Endorphins are endogenous opioid neuropeptides produced by the pituitary gland and the hypothalamus in response to stress, pain, or exercise.
They bind to mu ($\mu$) opioid receptors in the brain to inhibit pain transmission.
Step 2: Detailed Explanation:
Let us analyze the chemical structure of $\beta$-endorphin:
$\beta$-endorphin is synthesized as a large precursor protein called pro-opiomelanocortin (POMC).
POMC undergoes tissue-specific post-translational cleavage by prohormone convertases to yield several bioactive molecules, including ACTH, melanocyte-stimulating hormone ($\alpha$-MSH), and $\beta$-lipotropin.
The cleavage of $\beta$-lipotropin yields $\beta$-endorphin, which is a polypeptide consisting of exactly 31 amino acid residues arranged in a specific sequence:
\[ \text{Tyr-Gly-Gly-Phe-Met-Thr-Ser-Glu-Lys-Ser-Gln-Thr-Pro-Leu-Val-Thr-Leu-Phe-Lys-Asn-Ala-Ile-Ile-Lys-Asn-Ala-His-Lys-Lys-Gly-Gln} \]
Because it is composed of a long chain of amino acids linked together by peptide bonds, its chemical nature is strictly that of a polypeptide or protein.
It is not a lipid, steroid, or carbohydrate.
Step 3: Final Answer:
Thus, the chemical nature of $\beta$-endorphin is a polypeptide, which corresponds to option (C).