It seems probable to me that the answer to the question will be found by counting back from today through the Judaic calendar, down an unbroken cycle of days in the sequence Friday-Thursday-Wednesday-Tuesday-Monday-Sunday-Saturday.

Why the Judaic calendar?

There are three components to the question. The calendar must be:

• as old as possible (obviously)
• extant (still in use today, not one that has disappeared)
• unbroken (this is the part of the question that seems to me to be missed out by the other answers: that it includes "the specific cycle we're still on")

Antiquity of the calendar

The Judaic calendar has been around since the 6th century BC at least.

It has existed longer than the calendar that started with the Romans: in AD 351 (according to Wikipedia) "Emperor Constantine officially decreed a seven-day week" across the Roman Empire, that included a Sunday as a public holiday.

It has also existed longer than the Chinese and Indian calendars, according to the evidence available.

In current use

There could be other calendars, that ran for longer - but are no longer in use. At any rate, the Judaic calendar is very much in use.

Part of an unbroken cycle

The week is of defining importance in Judaism. It seems like something that Jewish culture would hold on to with some vigour. It's possible that the thread were broken at some point in Jewish history, and had to be restarted, but I have never heard of that.

On the other hand in Babylon and more recently in Roman times the frameworks of Jewish culture were at risk, so if the cycle were broken that would seem a likely time.

After the Romans, the dispersal of Jews actually makes it easier to keep the calendar going - distributed calendar-keeping, rather than being at the mercy of a single, dominant other system. For example, if it turned out that Jews in North Africa and Eastern Europe thought a different day was the first day of the week upon encountering each other after a long period, that would tell us that a thread had been broken - but I haven't heard of that either.

Best guess

So my best guess would be to look for evidence in Jewish history, and that the unbroken cycle might go back beyond the Romans, but that the start of the thread is unknown - we don't have a Constantine in this case to give us a convenient decree.

But, the Babylonians made lots of decrees, and it seems possible that they adopted the Jewish calendar, so perhaps a firmer answer can be found there.

Complication

One complication is that "our" (for some value of "our", I realise that this doesn't apply universally) days start at midnight, while others use sunrise or sunset. But we could probably mostly agree at midday.

Those who believe in the strict truth of the book of the Bible (I do not myself) would of course say it goes back to God's creation of the world, see Genesis Chapters 1 and 2

Most others say the week comes from Ancient Babylon, known directly from at least 500 BC but presumably older, if the Jewish religion absorbed it during the Captivity in Babylon which is thought to have ended 538 BC see:

https://www.timeanddate.com/calendar/days/7-days-week.html

More information in Gerard Clarke's book 'Heirs to Lost Kingdoms'

The 7 day week's origins appear to be based on Babylonian Astrology, which knew 7 'planets' (i.e. wanderers in the sky, as opposed to the fixed constellations): Sun, Moon, Mercury, Mars, Venus, Jupiter and Saturn. Each was thought to 'rule' the first hour of a day of the week. This made each day propitious for some activities but not for others, although there was not originally a Sabbath or a Weekend.

Hence why several religions that later developed in the Middle East have one day of a 7 day week as their Holy Day, but possibly also why there is no consensus as to which day it is: Wednesday for Yazidis, Thursday for Druze, Friday for Muslims, Saturday for Jews, Sunday for Christians. [For some reason no religion seems to like Mondays!?]

This took longer to spread to Europe. The 1st Century BC Roman writer and politician Cicero complained that the Jews were 'lazy' because they refused to work 1 day in 7.

His contemporary the Roman general Pompey had an advantage attacking Jerusalem in 63 BC because the Roman besiegers worked 7 days a week to build up ramps from which to attack the city walls while the pious Jewish defenders would only work to knock them down 6 days a week and stopped for the Sabbath.

Timothy is somewhat correct with identifying Babylon, however the Babylonians only borrowed the system. Most of our time keeping dates all the way back to ancient Sumeria (2600BC-ish?) and is mentioned in the epic of Gilgamesh.

Earliest we can trace it is between 2600BC and 3000BC.