I have an esp8266 hooked up to a solar inverter. it used to have problems with not getting enough power so i hooked up a power bank between the inverter and the esp. if the power from solar gets out, the power bank (10k mah on paper, must be less) can run it for around 36 hours on full charge.
Finding a powerbank that doesn't auto shut off with small power draws from a esp/microcontroller is hit or miss, it's not something that is advertised and the threshold can vary wildly. There are workarounds but they can get a little complex if you don't want them to just burn a bunch of excess power.
I've also had bad luck with power banks that can survive outside more than a little while. Not moisture, you can seal that out, but heat and cold seems to kill the batteries.
