Carbon dating only works on very recent things (up to a few tens of thousands of years). For older things like the earth and these fossils, slower-ticking radiometric clocks (like Uranium-Lead clocks) must be used. There was a great COSMOS episode discussing the history of it [1] but the fascinating math and science behind it is better explained for the layperson in [2].
Well, I meant very recent in the geological sense ;)
Good calibration curves accounting for things like that in your link are definitely required to get modern-era results, and with a 5,730 year half-life, it's difficult to even make meaningful measurements to throw onto the curve. So you're right.
The used U-Pb dating of zircons. They did not date the fossil directly, instead they dated other rocks in the same stratigraphic sequence (Isua supracrustal belt). The relevant sentences:
The stromatolite discovery locality (Extended Data Fig. 1) is within the hinge of an anticline cored by 3,709 ± 9-Myr-old andesitic metavolcanic rocks with locally preserved pillow structures and a maximum metamorphic temperature of ~550 °C17, 18. The pillowed metavolcanic rocks are overlain by bedded dolomite-rich metasedimentary rocks and in turn, by interlayered quartzites and metamorphosed banded iron formation that contain rare, small, high Th/U oscillatory-zoned volcano-sedimentary zircons with ages of 3,699 ± 12 and 3,691 ± 6 Myr3, 18.
That means rocks with zircons dated to 3709 Ma are interpreted to be below (older) and rocks with 3699 Ma zircons are interpreted as being above (younger).
Zircon is an excellent mineral for geochronology as it contains U and Th. Two U isotopes, 235U and 238U, plus 238Th all decay to Pb at different rates (half lives). Zircon forms by growing from magma, and does not contain any Pb when grows, as Pb does not fit into the crystal structure, so any Pb you measure in a zircon today is from radiogenic decay of U and Th. If the ratios of the individual U and Th isotopes and their respective Pb daughter products are all consistent with the same age, i.e. you get the same age when you back calculate to zero Pb for all three isotope systems, then you can be very confident in the date, hence the small uncertainties on the reported dates from the paper.
There are some very reliable ones. C14 is based on isotope ratios and is kind of crap, but uranium-lead dating is much less ambiguous because of chemical differences between uranium and lead.
According to that boundless.com link, "The half-life of carbon-14 is 5,730 years, so carbon dating is only relevant for dating fossils less than 60,000 years old."
But from the Wikipedia about radiocarbon method: "Radiocarbon dating is a radiometric dating method that uses (14C) to determine the age of carbonaceous materials up to about 60,000 years old."
