I was thinking about this in terms of building a product.
Even with that, a clock generally runs for months, even years, on a single battery. Not exactly phone territory, but might be feasible on the ESP8266, depending on how the oscillator behaves in deep sleep mode.
There are many dedicated real time clock chips, pre-designed for backup time-keeping and low power usage. Interfacing with one is going to be way faster/easier/better than trying to roll your own on a general purpose microcontroller, IMO.
TI's bq32000 is $0.55@1K, is 3.3V for easy interface to the ESP, and takes just over 1 microamp in backup power mode. That's just one that I quickly poked at.
(Side note: at this point, I'd think I'd want to be using an ESP32 in any new designs.)
Even with that, a clock generally runs for months, even years, on a single battery. Not exactly phone territory, but might be feasible on the ESP8266, depending on how the oscillator behaves in deep sleep mode.