Chennai International/domestic airport is the WORST new airport in India. Except for the arrivals areas, there is no other facility nearby to rest. The new international terminal is a good 10-15 walk away from the domestic, but they have golf carts to get you there. So much space has been wasted in this airport.
You have to get out to nearby hotels if you want a quick nap/shut eye. Do not try to get those retiring rooms in the airport, which are not in good shape at all. (personal experience). Please visit the Makemytrip.com counter in the arrivals and they will hook you up with a good 3-4 star hotels nearby(pallavaram area). Hotel Mahalaya is a good choice. The baggage drop/cloak room/storage area is a joke, just a shed outside the domestic arrivals. The cafes inside are ok, but if you want a hot meal, Sangeetha is really good, they are open 24 hours, closed only from 2-3am.
The toilets in Chennai int. are a mess, don't even try stepping in one. They stink and totally not cared for. you may actually catch some disease. Having flown into new terminals in BOM, BLR, HYD, chennai is by far the worst airport in terms of management, cleanliness, comfort, accessability,etc. Even the gate numbers are not right. And the airport staff always seem to be doing you a favor for everything. But the one shining spot is the airline staff(Indigo, Jet, BA, AI, all airlines) they go out of the way to make your few hours in chennai airport better. The sad part is none of these international airports have a STD/ISD phone booth where you can make calls from. For many international travelers, this is a handicap.