Hi there.

Yes,  AIIMS Nagpur  does have a gym and a sports ground. And also many more facilities besides these.

• Nagpur's All India Institute of Medical Sciences is sprawled across 150 acres.The air-conditioned library (G+2 Building) is fully centralised and has a total covered area of 2546 square metres. E-learning is supported by the Wi-Fi-enabled library complex. A group study room, a mini-conference room, and a computer facility with internet access are also available in the AIIMS Nagpur library complex.
• The AIIMS, MIHAN Nagpur OPD has routine investigative facilities including as pathology, biochemistry, and microbiology, as well as radiography.
• The AIIMS Nagpur hostel provides contemporary accommodations for undergraduate and postgraduate students. The undergraduate hostel is a thirteen-story structure with a total size of approximately 4939 square metres.

http://aiimsnagpur.edu.in/node/6