We don't really know
The recent development of our current digital environment (commercial use of the internet dates back to roughly 1980 - which also coincides for the approximate start of home computing), means that we haven't really had an opportunity to test their long-term viability (essentially we can't even test the digital data standards & storage methods for more than about 35 years because they simply haven't been around longer than that).
But, currently all of the standard storage mechanisms that we use today are only expected to remain viable for from a few years to a few decades (this includes so-called archival media like optical disks and data tapes).
So far we've never encountered a need for extremely long duration archiving of data, so no one has ever bothered to design a system to work for that situation. If the scientists and engineers in your story had a few years of warning, they could probably develop something that would work.
I do not know how it would look but, based upon experience with various methods, I can guess.
But maybe we can guess
The F-15 originally was built with "primitive" magnetic core memory. This type of memory is non-volatile and highly resistant to EMP and other things (like cosmic rays) that can damage the data stored in modern memory. However, it is much slower and bulkier than modern memory.
Magnetic Core Memory Durability
Core memory is non-volatile storage—it can retain its contents
indefinitely without power. It is also relatively unaffected by EMP
and radiation. These were important advantages for some applications
like first-generation industrial programmable controllers, military
installations and vehicles like fighter aircraft, as well as
spacecraft, and led to core being used for a number of years after
availability of semiconductor MOS memory (see also MOSFET). For
example, the Space Shuttle flight computers initially used core
memory, which preserved the contents of memory even through the
Challenger's disintegration and subsequent plunge into the sea in
1986.
POST
I imagine your computer's bootstrap would be composed of similar bulky but reliable and non-volatile memory. This basic bootstrap functionality, perhaps similar to your computer's POST (power on self-test), would ensure important portions of the computer still worked and would then (slowly) load the actual operating system for the device.
Flag failed components/use good ones
Because I would expect many of the bits of the computer to have degraded enough to be unusable, the overall system probably would provide massive redundancy for each critical component. As the POST operations encounter failing components, it'd automatically switch to testing the next redundant component in lines. Since the POST operations would likely be fairly elementary, the overall system would likely flag the "failed" component for re-evaluation by the full-up OS once the boot cycle completed. A more thorough mapping of the essential components (e.g. CPUs might reveal just certain portions of the chip failed and that the CPU was otherwise OK). The OS would use this map of its redundant components to ensure it could keep operating as long as a complete set of essential functions remained operational.
After boot up cycle
This computer system would probably fall back on a bank of relatively modern memory chips for actual operations after the initial bootstrap. It'd be up to the original POST operations to initially determine which banks of modern memory were still viable and then (like with the CPU), a more sophisticated utility in the OS would perform a more thorough mapping of the memory to see how much of it remained usable.
Data recovery
After the basic OS and self-check programming began operating, the computer would begin to activate its many RAID (redundant array of independent disks) like data storage systems. The "drives" in the system would be special low density (and probably solid state) memory drives. The RAID system would verify the bit states across multiple drives and slowly reconstitute any damage data in the storage systems.
Slow and reliable (tortoise) performance
In your scenario, the primary goal of the hardware would be reliability and data redundancy so the storage arrays for your data would be quite large and probably not all that fast. A set of fast "working" hard drive storage might be provided for daily operations.
The time it took for the RAID like systems to perform the data validation checks and/or rebuild damaged sections could be quite lengthy (days, weeks, or substantially longer - depending upon the speed of the devices and amount of data we're discussing). From a dramatic perspective this might allow the author to perform a variety of reveals through the course of the book as different sections of the data storage are flagged as "ready for use", loaded into the faster systems, and made available to the characters in the story.
If the data reconstruction was imperfect it might allow the computer to provide false information too...
All good things come to an end
All hardware eventually fails.
Meaning even if your computer booted perfectly upon the application of power, mechanical hard drives fail, solid state drives fail, memory fails, etc. Your computer that survived the centuries would eventually wear out and stop working. It should make that point to the inheritors of the system as soon as possible.
And another thing
Richard Feynman sponsored some prizes to groups that could write data ("There's Plenty of Room at the Bottom", in conventional analog form, in the highest density. For instance trying to print the Encyclopedia Britannica on the head of a pin. The only thing you'd need to read the data is a really good microscope. This sort of data's shelf life is potentially MUCH higher than that of digitally stored data and you wouldn't have to worry about computer interoperability and changes in encoding standards as a condition of data retrieval!
"There's Plenty of Room at the Bottom" was a lecture given by
physicist Richard Feynman at an American Physical Society meeting at
Caltech on December 29, 1959.1 Feynman considered the possibility of
direct manipulation of individual atoms as a more powerful form of
synthetic chemistry than those used at the time. The talk went
unnoticed and it didn't inspire the conceptual beginnings of the
field. In the 1990s it was rediscovered and publicised as a seminal
event in the field, probably to boost the history of nanotechnology
with Feynman's reputation.
...
At the meeting, Feynman concluded his talk with two challenges, and he
offered a prize of $1000 for the first individuals to solve each one.
The first challenge involved the construction of a tiny motor, which,
to Feynman's surprise, was achieved by November 1960 by William
McLellan, a meticulous craftsman, using conventional tools. The motor
met the conditions, but did not advance the art. The second challenge
involved the possibility of scaling down letters small enough so as to
be able to fit the entire Encyclopædia Britannica on the head of a
pin, by writing the information from a book page on a surface 1/25,000
smaller in linear scale. In 1985, Tom Newman, a Stanford graduate
student, successfully reduced the first paragraph of A Tale of Two
Cities by 1/25,000, and collected the second Feynman prize.
