Here are the specifications of the SG90 (from FEETECH (Fitec) FS90 9G Mini Servo with Accessories)
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.
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.