For battery power I would not use a Nano. The Nano runs at 16MHz, has a voltage regulator, a USB to TTL chip and some LEDs that all suck up power. Add a 3.3V regulator to get the power for the rf24 and that just adds more (the Nano 3.3V regulator will not supply sufficient current for a rf24 while transmitting).
A stand alone* mega328 (the processor in the Uno or equivalent) running at 8MHz will work from 2 AA batteries (3V nominal). The rf24 spec says that it will work from 1.9V to 3.6V so will be happy at that voltage, too.
Vastly extend battery life by sleeping the radio and using the low power techniques shown in Nick Gammon's page on power saving and stand alone processors.
*No voltage regulator(s), USB to TTL, LEDs. See the Nick Gammon link.