Arduino Nana + Batteries + GSM Module - possible or not?

A typical AA battery is around 2500 mah , and the battery voltage is 1.2 V if its a rechargeable
or 1.5 V if its a primary battery.
To run an Arduino you need either 5V regulated , or higher if you want to use the onboard regulator
on the Arduino board.
So you need at least 4 1.5 V AA batteries in series , or 5 1.2 V in series.
Even thats a bit marginal for the onboard voltage regulator.
With batteries in series , the same current flows through them all, so the mah capacity of the series
batteries is the same as one of them .