Before delving into the details of chemistry-less hours-lasting storage, first, the clock itself.
Let's make at least this thing simple. I am going to use a STM32L476-DISCO board. This is based on the STM32L476VG microcontroller, which is member of a low-power 'L4 mcu family by ST Microelectronic. The chip contains both a built-in RTC (with an oscillator for the 32.768kHz crystal) and a segment-LCD controller, making it a suitable choice for building a clock. The DISCO board, besides other things, features a 6 x 14-segment LCD, an on-board ST-Link debugger, and also an auxiliary microcontroller with circuitry for measuring the main mcu's current consumption.
There's one more thing making the 'L476 very suitable for our project: the RTC has its own power domain (intended to be powered by a battery, hence the respective pin is named VBAT), and it isolates itself from the main power domain automatically when the main power drops below the power monitor's threshold voltage.
Of course, there are drawbacks, too. The RTC alone, running with the 32768kHz crystal, consumes around 500nA (the exact figure is voltage dependent), which - while in itself impressive and perfectly suitable for all usual practical purposes - is not exactly the lowest current which could be found on the market. As mentioned in the "root" project, there is a commercially available integrated RTC-oscillator-crystal, with order of magnitude less current consumption, at around 40nA.
So, keeping things simple for now, let's just try to run a simple clock on the 'L476, using its RTC; trying to maintain backup for its relatively large power consumption for maybe one or two hours - and if we succeed in this, there's hope that using a less power-hungry RTC may result in the desired night-long endurance.