Car Sales India breakdown cover service is available in most parts of the country. They have two packages, GOLD(Rs 1250/-), and PLATINUM(Rs. 2500/-).

Not sure of about a AAA equivalent, but if you have new car, most manufacturers provide this kind of a service through a 24 hour hotline. Again, no idea if this service is available after say the car is 2-3 years old (and out of warranty).

Check this recent announcement from Hyundai. It says free roadside assistance for 2 years and available to older cars for 'some charges' http://hyundai.co.in/presspop.asp?newsID=496&newsTitle=Hyundai%20unveils%20customer%20friendly%20initiatives

Most local car dealers provide a 'on-call' service for emergencies.
I have used the Deccan Honda service a couple of times
(in one extreme case this was for a dead-battery too).
They will come, probably fast, and usually charge you high for the service.

Talk with your car dealer to check this out.
These days some dealers also provide an AMC service (think of this like an insurance).