Storing 14+ days worth of <everything> is going to mean massive amounts of storage.
...which is why you want to be extra careful in how mass-efficiently you're storing it.
A tank of liquid oxygen masses only a few percent of its contents, and its contents store more chemical energy per mass than a lithium battery. Later on, insulated tanks are vastly easier to produce from lunar materials vs batteries. These are all nice leverage multipliers.
Nuclear is a popular notion, but the real-world economics don't favor it. Effectiveness beats elegance.[0]
Making liquid oxygen (as opposed to gaseous oxygen) is not a terribly efficient process.
You have to do the trade between liquefaction and storing oxygen as a pressurized gas. Reusing a single Starship's propellant tanks, for example, at 6 bar would hold ~14 metric tons of oxygen, enabling a continuous lunar population of over 1,000 people.
The compression or liquefaction energy isn't a show-stopper necessarily, it just works into the total energy efficiency of the storage system. Compression is nice because practical isothermal compressors are now being demonstrated.[0]
Both liquefaction and compression require cooling. I do think it probably makes sense to do the thermal storage trick in reverse also, storing up a bank of "cold" at night to be used during the day. The thermal radiator panels still provide (derated) cooling in the day, but the thermal storage pool time-shifts the natural oversupply of cooling power at night.
[0] https://newatlas.com/good-thinking/carnot-centrifugal-air-co...
Care to elaborate? As an interested layperson I always thought the major factor for losses in open air was radiation into space. As such an atmosphere's thermal mass would be benfitial, no?
I think part of the solution will involve using "regular stuff" (which can be sourced from lunar atoms) like batteries.
Don't use lithium batteries to continuously power an oxygen generator. Instead, have an enlarged oxygen generator and store half the flow for use at night.
Don't use lithium batteries to power a heater at night, instead "pre-charge" heat in a dirt thermal mass during the day.
Don't use lithium batteries to power a continuous CO2 scrubber, instead use a liquid amine scrubber and recharge the fluid tanks during the day.
Etc etc
Obviously you need to do in-depth tradeoff math for each case, but my napkins say "the system is the battery" wins in many cases.