Answer
About 1,000 requests in flight. In review, anything from 667 to 1,500 counts as right.
The worked estimate
Little's law: in flight = arrival rate × time in the system. 5,000/s × 0.2 s = 1,000 requests in flight. If each holds a thread or a database connection, that is the pool size you need, and it doubles if latency doubles.
Numbers: Numbers to know.