While no one has ever turned off the sun, there are some historical reference points.
Volcanic Eruption Data
Following the May 20, 1883 eruption of the Krakatoa volcano which spewed ash into the air that slightly reduced the amount of sunlight reaching the Earth:
Average global temperatures fell by as much as 1.2 degrees Celsius in
the year following the eruption. Weather patterns continued to be
chaotic for years and temperatures did not return to normal until
1888.
Mount Tambora's eruption on April 10, 1815 (also in Indonesia) was the most powerful in recorded history:
The eruption caused global climate anomalies that included the
phenomenon known as "volcanic winter": 1816 became known as the "Year
Without a Summer" because of the effect on North American and European
weather. Crops failed and livestock died in much of the Northern
Hemisphere, resulting in the worst famine of the 19th century.
In particular:
The 1815 eruption released sulfur dioxide (SO2) into the stratosphere,
causing a global climate anomaly. Different methods have estimated the
ejected sulphur mass during the eruption: the petrological method; an
optical depth measurement based on anatomical observations; and the
polar ice core sulfate concentration method, using cores from
Greenland and Antarctica. The figures vary depending on the method,
ranging from 10 to 120 million tonnes.
In the spring and summer of 1815, a persistent "dry fog" was observed
in the northeastern United States. The fog reddened and dimmed the
sunlight, such that sunspots were visible to the naked eye. Neither
wind nor rainfall dispersed the "fog". It was identified as a
stratospheric sulfate aerosol veil.[10] In summer 1816, countries in
the Northern Hemisphere suffered extreme weather conditions, dubbed
the Year Without a Summer. Average global temperatures decreased about
0.4–0.7 °C (0.7–1.3 °F),4 enough to cause significant agricultural problems around the globe. On 4 June 1816, frosts were reported in the
upper elevations of New Hampshire, Maine, Vermont and northern New
York. On 6 June 1816, snow fell in Albany, New York, and Dennysville,
Maine.[10] Such conditions occurred for at least three months and
ruined most agricultural crops in North America. Canada experienced
extreme cold during that summer. Snow 30 cm (12 in) deep accumulated
near Quebec City from 6 to 10 June 1816.
The second-coldest year in the Northern Hemisphere since c.1400 was
1816, and the 1810s are the coldest decade on record, a result of
Tambora's 1815 eruption and another possible VEI 7 eruption that took
place in late 1808 (see sulfate concentration figure from ice core
data). The surface temperature anomalies during the summer of 1816,
1817, and 1818 were −0.51 °C (−0.92 °F), −0.44 °C (−0.79 °F) and −0.29
°C (−0.52 °F), respectively.7 As well as a cooler summer, parts of
Europe experienced a stormier winter.
This climate anomaly has been blamed for the severity of typhus
epidemics in southeast Europe and the eastern Mediterranean between
1816 and 1819. The climate changes disrupted the Indian monsoons,
caused three failed harvests and famine contributing to the spread of
a new strain of cholera originating in Bengal in 1816. Many livestock
died in New England during the winter of 1816–1817. Cool temperatures
and heavy rains resulted in failed harvests in Britain and Ireland.
Families in Wales travelled long distances as refugees, begging for
food. Famine was prevalent in north and southwest Ireland, following
the failure of wheat, oat, and potato harvests. The crisis was severe
in Germany, where food prices rose sharply and demonstrations in front
of grain markets and bakeries, followed by riots, arson, and looting,
took place in many European cities. It was the worst famine of the
19th century.
A volcanic eruption in 1257 CE in Lombok, Indonesia probably caused the Little Age Age. What happened half the world away (where all of the effects were from loss of sunlight rathe than direct impacts of the volcano):
Reports in 1258 recount the presence of a dry fog, giving the
impression of a persistent cloud cover to contemporary observers.
Medieval chronicles say that in 1258 the summer was cold and rainy,
resulting in floods and bad harvests, with cold lasting from February
to June. In both Europe and the Middle East, changes in atmospheric
colours, storms, cold and severe weather were reported in 1258-1259.
In Europe, too much rain damaged crops and caused famines followed by
epidemics. Reports of the effects of the eruption, including failure
of crops and famine as well as weather changes, exist for northwest
Europe. Crop failures, and a famine in London have been linked to this
event. Witnesses reported a death toll of 15,000 to 20,000 in London.
Matthew Paris of St Alban retells how until mid-August in 1258, the
weather alternated between cold and strong rain, causing high
mortality.
Swollen and rotting in groups of five or six, the dead lay abandoned
in pigsties, on dunghills, and in the muddy streets.
— Matthew Paris, chronicler of St. Albans,
The resulting famine was severe enough that grain was imported from
Germany and Holland. The price for corn increased in Britain,
France and Italy. Outbreaks of disease occurred during this time in
the Middle East and England. Problems were also recorded in China,
Japan and Korea. Other effects of the volcanic eruption include a
lunar eclipse in May 1258, where the Moon was completely darkened.
With and after winter 1258-1259, exceptional weathers were reported
less commonly, but the winter 1260-1261 was very severe in Iceland,
Italy and elsewhere.
The K-Pg extinction event
The most pertinent precedent is the K-Pg extinction event (formerly known as the K-T extinction event) 66 million years ago that killed off the dinosaurs clearing the way for mammals:
[A] 10-to-15-kilometre (6.2 to 9.3 mi) space rock hurtled into Earth
at Chicxulub on Mexico's Yucatán Peninsula. The collision would have
released the same energy as 100 teratonnes of TNT (420 ZJ), over a
billion times the energy of the atomic bombings of Hiroshima and
Nagasaki.
The consequences of the Chicxulub impact were of global extent. Some
of these phenomena were brief occurrences that immediately followed
the impact, but there were also long-term geochemical and climatic
disruptions that were catastrophic to the ecology. . . . the impact
would have inhibited photosynthesis by creating a dust cloud that
blocked sunlight for up to a year. Further, the asteroid struck a
region of sulfur-rich carbonate rock, much of which was vaporized,
thereby injecting sulfuric acid aerosols into the stratosphere, which
might have reduced sunlight reaching the Earth's surface by more than
50%, and would have caused rain and ocean water to become
acidic. The acidification of the oceans would kill many
organisms that build shells from calcium carbonate. At Brazos section,
the paleo-sea surface temperature dropped as much as 7℃ for decades
after the impact. It would take at least ten years for such
aerosols to dissipate, and would account for the extinction of plants
and phytoplankton, and of organisms dependent on them (including
predatory animals as well as herbivores). Some creatures whose food
chains were based on detritus would have a reasonable chance of
survival.
New studies show that this caused mass extinctions at levels previously unrealized:
Extinction rates are markedly higher than previously estimated: of 59
species, four survived (93% species extinction, 86% of genera).
The "good news" is that even a year with an at least 50% reduction in solar energy penetration wasn't enough to cause all life on Earth to go extinct, although the vast majority of life on Earth including virtually all "megafauna" did.
The "bad news" is that a far shorter deprivation of sunlight, particularly if it was more complete than the K-T event, would still be a big problem.
The 7 degrees of temperature drop from a 50% reduction in sunlight for on year might be particularly useful in estimating the impact on the climate from a 100% reduction in sunlight for some period of time shorter than that. After all, we know from the lower impact volcanic events that even a 1-2 degree temperature drop for a year or so is a big thing.
The Heat Budget Of Earth
We know from basic thermodynamics that heat loss is a function of temperature difference. Heat radiated from the parts of the Earth where people live is mediated by the atmosphere which is between us and empty space. Empty space is about 3 degrees K.
But, the top of the atmosphere, even after the K-Pg extinction event, would still have been heated by the sun without interruption - less of that heat would have made it to the surface of the Earth, but the heat on the top of the clouds would still have slowed radiation of heat to outer space. To evaluate that you need a grip on the Earth's heat budget in normal times:
To quantify Earth's heat budget or heat balance, let the insolation
received at the top of the atmosphere be 100 units, as shown in the
accompanying illustration. Called the albedo of Earth, around 35 units
are reflected back to space: 27 from the top of clouds, 2 from snow
and ice-covered areas, and 6 by other parts of the atmosphere. The 65
remaining units are absorbed: 14 within the atmosphere and 51 by the
Earth’s surface. These 51 units are radiated to space in the form of
terrestrial radiation: 17 directly radiated to space and 34 absorbed
by the atmosphere (19 through latent heat of condensation, 9 via
convection and turbulence, and 6 directly absorbed). The 48 units
absorbed by the atmosphere (34 units from terrestrial radiation and 14
from insolation) are finally radiated back to space. These 65 units
(17 from the ground and 48 from the atmosphere) balance the 65 units
absorbed from the sun; thereby demonstrating no net gain of energy by
the Earth.
So, about a third of the solar energy normally received on Earth immediately goes into space through radiation, and two-thirds of the solar energy normally received on Earth is absorbed by the atmosphere and ultimately ends up in space, but more slowly, preventing the surface from losing energy all that much faster.
The Precedent Of Nighttime
It would also help to know the typical differences between night and day temperatures:
As solar energy strikes the earth’s surface each morning, a shallow
1–3-centimetre (0.39–1.18 in) layer of air directly above the ground
is heated by conduction. Heat exchange between this shallow layer of
warm air and the cooler air above is very inefficient. On a warm
summer’s day, for example, air temperatures may vary by 16.5 °C (30
°F) from just above the ground to waist height. Incoming solar
radiation exceeds outgoing heat energy for many hours after noon and
equilibrium is usually reached from 3–5 p.m. but this may be affected
by a variety of different things such as large bodies of water, soil
type and cover, wind, cloud cover/water vapor, and moisture on the
ground.
Diurnal temperature variations are greatest very near the earth’s
surface.
High desert areas typically have the greatest diurnal temperature
variations. Low lying, humid areas typically have the least. This
explains why an area like the Snake River Plain can have high
temperatures of 38 °C (100 °F) during a summer day, and then have lows
of 5–10 °C (41–50 °F). At the same time, Washington D.C., which is
much more humid, has temperature variations of only 8 °C (14 °F);
urban Hong Kong has a diurnal temperature range of little more than 4
°C (7.2 °F).
Averaging the high and the low provides about 18 °C of average temperature difference between day and night.
The temperature loss from a single night without the Sun would probably triple in 36 hours without sunlight, so even the best buffered areas might lose 24 °C and the least buffered areas, like deserts, would lose much more, perhaps 54 °C or more, which would be enough to turn hot deserts into frozen deserts. (Cooler temperatures would also shut down evaporation leading to extreme aridity in every place that depends on rain once the exiting moisture in the air was exhausted.)
This suggests that ambient temperature drops will wreck havoc in a matter of hours or days by exposing people and livestock and plants to temperatures that are cold enough to kill them from exposure, while loss of photosynthesis leading to famines and the like (given food stores on hand) might take months; after all, lots of places have months of winter when nothing grows already.
Conclusion
Many people, animals and plants would die from exposure to cold in even a single day entirely without any Sun with a night before and after the period without the Sun (as opposed to merely being obscured by particulates in the atmosphere).
A "safe" period during which the Sun could be turned off would be significantly less than 12 hours.
An entire 12 hours without Sun deprives every place on Earth from Sun for 24 to 36 hours, while 4-6 hours would lead to Sun deprivation for no more than 18 hours continuously in the worst hit areas, and would have less continuous deprivation in many places. Breaking up a continuous period completely without the Sun for even a few hours makes a huge difference relative to having the Sun turn off just before sunrise and stay off until what would ordinarily be sunset in terms of planetary and local temperature drops.
On the other hand, Paige Ksnak's estimate of one degree per minute is probably too high, mostly because the atmosphere is still a blanket that would slow heat loss and buffer the temperature of the Earth until it lost all of its heat, which we know takes more than twelve hours without exposure to the Sun to happen because parts of the planet experience that much time without Sun every day. Indeed, even just six or seven hours of Sun followed by twenty-one hours of dark, while making it much cooler, doesn't cause catastrophe if you're ready for it.
So, I suspect that the practical limit would be driven by temperature losses from sustained continuous periods of time without the Sun and would be on the order of 4-6 hours tops before severe consequences and mass death followed.
