Historically, a lot of "mothballed" ships were put into that status due to the war that they were ordered for ending. This happened with a lot of US Navy ships at the end of WW2 - lots of carriers were completed and did not enter normal service because the war they were ordered for had ended. This did give the USN a bunch of unused ships that could be used for research and experimentation (such as the trials with angled flight decks).
There's also the status of "in reserve" which can be considered - even during WW2 itself, some ships were in reserve, with reduced crews, because they weren't needed. Many of the UK's battleships were "in reserve" in 1944, with only the modern ships in active service, because the older ships just were not needed, and the crews were more usefully used on other ships.
Trained manpower was a constraint, that factored in to the decisions to mothball ships, as well as to keep others in reduced readiness in reserve.
Aurora doesn't have a reserve/reduced readiness mechanic, as well as not having a mothball mechanic.
I have a bunch of ships in my current game, that are largely idle at the moment, but... most of them are commercial ships so the maintenance isn't a factor.
Many of the constraints that Aurora has on fleet size, are fairly easily dealt with by the player. You can get more officers and crew by expanding academies, the wealth and maintenance requirements are also not greatly limiting (other than the gallicite question).
Officers, crew, MSP, wealth, are the limiting factors in Aurora for fleet size, and they're not usually that limiting.
Many of the other things that affected ship design and fleet size historically are less of an issue in Aurora. In the case of the UK, there were several "War Emergency Programme" designs for ships, that could be built rapidly, and some of the other wartime designs were not built to the same standard as peacetime constructions. One of the merchant ship designs was built to use older types of engines - the thinking behind that was that a middle-aged reservist who hadn't been to sea in 20 years should be able to recognise every piece of machinery from when they were an apprentice.
Aurora's ability for shipyards to build from components functions pretty well as a WEP equivalent, and there's no difference between ships built in peacetime and ships built in wartime.
So I'm really not sure of why there would need to be a "mothball", or "reserve" mechanic for Aurora.
Now then, for a mechanism, I think that a substantial amount of the mechanics (in terms of coding) already exist in Aurora - the crew training level, conscript crew, and the "exit overhaul" function.
My preference would be a "mothball" facility, similar to a maintenance facility - it costs wealth and population to run, and can only support X tonnage of ships.
Mothballed ships require no crew, no officers, and won't explode whilst in storage.
Getting the ship out of mothballs though...
it starts at 0% readiness (just like an early exit from overhaul) and takes time to run up to 100% readiness (affected by captain/chief engineer's skills).
And it starts with the lowest crew grade (crewed by ageing reservists, and people who aren't familiar with old technology).
During the time it takes to run up to 100% readiness, it suffers wear and maintenance issues similar to ships undergoing fleet drills (stress of reactivating machinery from cold etc).
So whilst it was mothballed, the ship wasn't costing you anything, but in the say 100 days to run up to full working order, it's going to consume a lot more MSP, and be in a fairly worn state (possibly requiring immediate overhaul) by the time it is ready for service, and would consume even more MSP to train the crew up to a higher standard.
So you have the investment in minerals & population to build mothball facilities, the wealth cost to operate them, which are all things that you could be using on other more productive planetary installations.
And bringing ships out of mothball is expensive, and you end up with a worn ship with a relatively untrained crew, that needs further overhaul and training, before it equals a ship of the same class that wasn't put into mothballs and remained active.
Which I think counters "build to mothballs" to at least some extent.