From this answer to Power supply circuit for powering sg90 servo. How to handle high current requirements?, lets assume each servo (best case) needs 250 mA. I am still trying to locate a link to a datasheet with the current ratings stated
The Nano requires 280 mA, let's say (see this answer to What is the maximum power consumption of the Arduino Nano 3.0?)
Using a single 5V supply to power both servos and Arduino. The current rating needs to need at least:
I = (9 x 0.25) + 0.28
I = 2.53 A
Or power rated
P = I x V = 2.53 x 5
P = 12.65 W
You probably want more than that, so get a 5 V 5 A supply to be on the safe side. So, no, a battery will not suffice.