My Mazda MPV suddenly won’t start. When I turn the car on and try using the electronic locks, that “locking” sound sounds very weak and the car is unable to lock or unlock itself. I tried jumping it, and I took the battery to an auto shop which said that battery was fine. I recall that last time I drove the car, the O/D light was blinking, which I found odd. So, what’s wrong with my car? 
Clean and check battery cable connections, specially the ground bolt on the engine block.
A blinking O/D light indicates an electronic problem with your transmission, and this is not something to take lightly, as an electronic trans problem that is ignored can morph into a much more expensive mechanical problem with the transmission.
That being said, I am unsure of how a trans problem would preclude being able to start the engine.
I think that, at this point, you have little recourse other than having the car towed to a competent mechanic.
If this mystery-vintage Mazda is still under warranty, obviously it should be towed to a Mazda dealer.
If the warranty has lapsed, a competent independent mechanic (NOT a chain like Midas, Meineke, Monro, Sears, Pep Boys, AAMCO, etc) should be able to diagnose and fix the problem(s).
I bet Barky is right on point. The OD light may very well be related to the system starving for the right battery voltage.