That sounds scary—if you miss "over the next ten thousand years."
the news story is based on is webbed. It gives an estimate of the consequences of burning all of Earth's fossil fuel over the next few centuries. The conclusion is that "Antarctica is projected to become almost ice-free with an average contribution to sea-level rise exceeding 3 m per century during the first millennium." Figure 1d from the article shows a rapid rise, close to forty meters in the first thousand years, gradually tapering off thereafter.
Three meters a century is considerably faster than the current rate of rise—not surprising since it would be driven by a CO2 concentration about eight times the current level. One way of getting a feel for how serious it would be is to convert it into a rate at which coastlines move inward, assuming no diking. The rule of thumb for that is about a hundred meters shift for every meter of rise. So three meters per century of SLR implies coastlines shifting in by about three hundred meters a century, more in some places, less in others, but well short of the catastrophe implied by "coastal cities deep underwater."
Another way of looking at it is in terms of what could be done via diking. The lowest city in the Netherlands, a country with centuries of experience protecting land below sea level, is more than six meters below sea level. With two centuries to do it in, I expect New York or Shanghai could match that.
Another conclusion of the article is that the Antarctic would end up almost ice free, but it is not clear why that would be a bad thing. We are currently in an ice age, defined by the existence of ice on the poles. There have been earlier ice ages, but for most of the history of the Earth, including most of the period with living creatures and even most of the period with mammals, the poles have been free of ice.
Figure 1c shows the estimated effect on average global temperature—an increase over about a thousand years of a little over 10°C relative to the current value, followed by a gradual decrease thereafter. That looks like a much bigger problem than sea level rise—but how big? At the same time that the hotter parts of the Earth were becoming unihabitable, the colder parts—Antarctica and currently frozen parts of the Northern Hemisphere—would be becoming habitable. An accurate calculation of the net result would require more expertise than I have and more effort than I am willing to put into the project, but I can at least try a rough back of the envelope estimate.
Warming due to CO2 tends to be greater in cold times and places than in hot, because water vapor is also a greenhouse gas, the warmer it is the more water vapor is in the air, and the more of one greenhouse gas the less the effect of adding another. I do not know how large the difference would be for warming on the scale I am looking at. To simplify my calculations, I will assume that high temperatures in hot areas go up by five degrees, low temperatures in cold areas by fifteen—readers are welcome to recalculate the numbers with other assumptions.
How high do high temperatures have to be to make a place uninhabitable? India is a very hot place and densely populated. Looking at a list
of July high temperatures by city, I observe a high of 36°C in Tamil Nadu and Pondicherry. If that represents the upper bound for habitability, another five degrees would make any city currently above 31° uninhabitable. Counting cities, which is less work than looking up regional areas and adding them, I observe that twenty-four out of fifty-two cities are above that, so my very rough estimate is that the projected warming would make almost half of India uninhabitable by a thousand years hence.
Looking at a list of highest temperatures ever recorded by country, the figure for India is 50.6°. Algeria, Iran, Iraq, Kuwait, Mexico, Pakistan and Saudi Arabia equal or exceed that, and a few other countries come close. If I add up the area of India and all countries whose highest temperature is at least as high, I get a total of about 12.7 million square km. To allow for countries a little cooler than India, take it up to 14 million, then divide in half, since my very rough calculation suggests that only about half of India would become intolerably hot.
Very, very rough estimate: A five degree increase in maximum temperatures would make about seven million square km intolerably hot. What would we get in exchange?
Antarctica and Greenland would be ice free, for a total area of about sixteen million square km. How much of that would be warm enough to be habitable I do not know. Siberia is about thirteen million, Canada about ten million. Parts of both are presently habitable, but large parts are not. Similarly for Sweden, Norway, and Finland.
I conclude, from my very rough estimates, that the total habitable area of the Earth would almost certainly go up, not down. I leave to someone more ambitious the task of a more careful and precise calculation. A lot of people would have to move—but a thousand years is a very long time.
Why does all of this matter, given the difficulty of predicting anything a thousand years, even a hundred years, into the future? It matters because it suggests an upper bound to climate catastrophe, at least slow catastrophe. In the worst case, the long term result would be a physical world not strikingly worse for humans, possibly better, than the present world.
If the changes happened over decades there would be enormous human costs. If they took many centuries, probably not.