Step 1: Understanding the Concept:
Ancient Indian literature evolved over centuries, starting from linguistic foundations to complex poetry and religious compilations.
Step 2: Detailed Explanation:
1. Ashtadhyayi of Panini (A): This is the earliest work among these, dated to approximately the $5^{th}$ or $4^{th}$ century BCE. It established the rules of Sanskrit grammar.
2. Natyashastra of Bharata (D): This encyclopedic work on performing arts is generally dated between 200 BCE and 200 CE.
3. Compilation of the Puranas (C): While Puranic oral traditions are very old, their systematic compilation into the Sanskrit versions we know today largely took place from the $3^{rd}$ century CE onwards.
4. The works of Kalidasa (B): Kalidasa is the pinnacle of Classical Sanskrit literature, associated with the Gupta golden age (late $4^{th}$ to early $5^{th}$ century CE).
Step 3: Final Answer:
The correct chronological order is A (Panini) $\rightarrow$ D (Natyashastra) $\rightarrow$ C (Puranas) $\rightarrow$ B (Kalidasa).
This matches option (C).