Time keeping, commonly, is stored as a binary number that represents how many seconds have passed since midnight (UTC) on January 1st 1970. Since the year 10,000 isn't x seconds away from epoch (1970-01-01T00:00:00Z), where x is any factor of 2 (aka 2^x, where x is any integer), any discrepancies in the use of "year" as a 4 digit number vs a 5 digit number, are entirely a display issue (front end). The thing that does the actual processing, storing and evaluation of time, gives absolutely no fucks about what "year" it is, because the current datetime is a binary number representing the seconds since epoch.
Whether that is displayed to you correctly or not, doesn't matter in the slightest. The machine will function even if you see some weird shit, like the year being 99 100 because some lazy person decided to hard code it to show "99" as the first two digits, then take the current year, subtract 9900, and display whatever was left (so it would show the year 9999 as "99", and the year 10000 as year "100") so the date becomes 99 concatenated with the last two (now three) digits left over.
I get that it's a joke, but the joke isn't based on any technical understanding of how timekeeping works in technology.
The whole W2k thing was a bunch of fear mongering horse shit. For most systems, the year would have shown as "19-100", 1900, or simply "00" (or some variant thereof).
Edit: the image in the OP is also a depiction of me reading replies. I just can't even.
I don't think 10000 year is a problem. There is a real "year 2038 problem" that affects system storing unix time in signed int32, but it's mostly solved already. The next problem will be in year 33000 or something like that.
In other news, the colony Szinthar failed to update its software systems due to a lack of pregrammers and Techmancers. Signals received suggest there were no survivors.
oh just start at 0000 again, signate that as 10,000. Files didn't start until like 1979 anyways, and there can't be many left, and even if it is a problem, now you have 2000 years to not worry about it.
Awww shit, time to rewatch my favourite Jike Mudge movie starring Lon Rivingston; Space Office (9999).
Haha, I can't believe this guy has the job of manually changing all the dates on the company's database, this place sucks. I bet the past was way better.