The sun inexplicably ceases to give light. It's still there and continues to have a gravitational pull keeping us in orbit. The solar system continues to operate as normal, but the sun has gone dark and ceases to provide us with light or energy.
Could humans, with sufficiently advanced technology, find a way to survive in this new world?
It would be brutally difficult
So the difficulty is the sheer magnitude of how much energy the sun applies to the Earth. It's an astonishing 1–1.5kW/m2! More energy hits the roof of your house every day than it would take to power your entire neighborhood if fully harnessed (solar cells aren't quite that awesome yet).
I turn to my favorite chart in the world, Wikipedia's comparison of energy levels by orders of magnitude:
While this isn't the end of the story, this should point out an important reality: The entire fossil fuel and fission energy reserves of the Earth account for roughly a month's energy from the Sun. Something is going to have to give.
Truth be told, we don't need the energy from the Sun. We can be more efficient if we explore extreme technological solutions. However, nature is rather dependent on this abundant energy source. Any solution we come up with is going to have to be purely synthetic, with little to no help from nature.
Extreme cheating
Okay, look. If we go to the ultimate in extreme tech, and manage to convert mass directly into energy, we're pretty unaffected. Even the fusion of the sun cannot match the efficiency we would get here. Just Uzbekistan's share of the planet would be enough to keep us going for a long time ($448 978 \text{km}^2$ = 0.088% of the surface of earth = $4.7×10^{38}\text{J}$ = 86,000,000,000,000 years worth of sunlight energy). In fact, we'd probably immediately send interplanetary dumptrucks to the dead sun to harvest its hydrogen and helium for energy!
Less cheating
If we could do fusion ourselves, rather than relying on the sun, the Deuterium in the ocean contains $1.5×10^{31}\text{J}$ of energy. That's just over 2.7 million years of energy. Of course, like the extreme cheating case, we'd probably eventually send dumptrucks to harvest unfused hydrogen from the sun. It just wouldn't be quite as extremely efficient as if we had perfect matter conversion.
Even less cheating
If we were to build a giant space generator with massive electromagnets on the moon, we could slowly bleed off its kinetic energy, giving us just shy of 7000 years worth of sunlight's energy.
Not so much cheating
Bollocks.
Okay, so what if we can't get huge amounts of energy to work with? What can we do?
The biggest issue is going to be heat generation. Humans really don't function well in ultra-cold settings, and neither do our food sources. As a lower threshold, let's target freezing. Any colder than that, and hydroponics becomes difficult (we're going to become herbivores really quickly).
Earth is going to become a frigid ball very quickly. We're going to need to effectively create a solar blanket to keep from radiating ourselves into a cold hell. Doing this with an atmosphere as chaotic as Earth's will be hard; we'll probably have to take a jaunt to the moon, where we could create an atmosphere within a solar-blanket envelope. Done right, with several layers, and we might be able to avoid freezing.
Now for our energy needs. The average terrestrial human in 2008 uses $7.7×10^{13}\text{J}$, plus $3×10^{6}\text{J}$ worth of calories. Even with inefficiencies, the calorie intake is of minimal importance. However, the cost of getting fuel is going to be going up. The oceans will freeze, so we can't do underwater drilling. We're going to be dependent on the energy sources we can find (and can send up to the moon, where we stand a chance of not freezing to death).
In all, its a miserable existence, waiting to slowly freeze. We'd probably make it, because humans are extremely resilient. However, the cost of all the depression counseling will be sky high!
This would be survivable for a small number of humans with not much more technology than we currently have.
You'd need to go down, far enough under the planet to get insulation and maybe geothermal heating. Use geothermal energy and/or fission reactors to provide light and grow plants in massive underground hydroponics chambers.
The good news is that actually there are plenty of resources available, if oxygen is running low just send some robots up, scoop some up that will by lying on the surface, bring it inside, and melt it.
Life would be hard, cramped, and poor. Space and energy would both be at a premium. It would also be vulnerable, one earthquake could end everything. It would provide a seed from which to expand out slowly and where further scientific advances could be made though.
We could survive for thousands, maybe even millions, of years this way before the earth's core lost all its heat.
We wouldn't be the only survivors either as the deep oceans, especially near hydrothermal vents, would remain viable for a long time.
Most likely not.
With our current level of technology, we could not survive such an event.
We would need to provide light for getting around and to grow all of our food. We would also have to somehow keep ourselves, drinkable water, and the atmosphere from freezing.
With a significant amount of warning and some better technology than we have now, a very small group of people may be able to survive, perhaps even a few decades, in a bunker deep underground. If they built it to be sustained by the Earth's internal heat. It's unlikely they could survive very long though. They would eventually run out of a crucial supply or need a replacement part that required a global society to manufacture.
It would be a dark time for humanity, literally and figuratively.
The situation you describe is similar to the situation rogue planets are in.
http://en.wikipedia.org/wiki/Rogue_planet#Retention_of_heat_in_interstellar_space
That wiki claims that even liquid water on the surface of an Earth-sized planet is possible, though it will require 1000 times increased atmospheric pressure if I'm reading it right.
But even if the surface is not suitable for life, the Earth's core will still be hot for a long time after such event, so we can dig in and survive. Especially if we keep the Moon with us, as it will keep providing tidal heating.
humans, with sufficiently advanced technology
Sufficiently advanced technology is the best kind of technology, it allows you to do anything within the realm of physics so long as there is a need. As @CortAmmon stated, the Earth has plenty of mass, as well as the Sun to be able to convert mass to energy well beyond the amount of energy produced by the sun.
However, sufficiently advanced technology is not exactly the same as sufficiently advanced engineering, although they are both often mistaken for the same. We have the technology to levitate trains, strawberries and frogs, but we still do not have flying cars, even though the technology is here.
Your humans may have performed tests in labs to transform mass to energy, but they may still be years or decades from fielding the technology. I am quite confident in the that a mass to energy plant does not yet exist, as a single working prototype could easily outperform any and all other forms of energy production, and humanity may simply just shrug at the sun going dark.
What I would imagine would happen is that humanity as we know it today, including the ones in colonies on other planets or in space, especially them in fact, would cease to be, but there may be a deep underground bunker lab where scientists and engineers toil away to get the mass to energy reactor to work, and when it does, there will be so much energy, it could easily send a 21st century human society to post-scarcity, but the energy requirements of humans who perfect the mass to energy reactor would be much higher, but I digress
According to Fritz Leiber's short story A Pail of Air, small groups of people could survive after losing the heat and light of the sun -- even after the atmosphere freezes out as "snow".
Nuclear reactors could provide heat and electricity for a very long time. Underground bunkers (if dug deep enough) could become a place for people to live, heated and lit and growing food.
In A Pail of Air, one family even manages to survive for years in an insulated room where they keep a fire going constantly. Every day they scoop up another pail of frozen oxygen snow from outside to boil off on the fire. Here's a description of their room:
Let me tell you about the Nest. It's low and snug, just room for the four of us and our things. The floor is covered with thick woolly rugs. Three of the sides are blankets, and the blankets roofing it touch Pa's head. He tells me it's inside a much bigger room, but I've never seen the real walls or ceiling...
The fourth wall has blankets all over except around the fireplace, in which there is a fire that must never go out. It keeps us from freezing and does a lot more besides.
When you toss in, "with sufficiently advanced technology", I think the answer is "obviously yes".
I certainly haven't worked out the details but I strongly suspect we could do it with current technology. Dig into the surface fifty feet or so, so you have plenty of insulation. Then build a power plant that puts out sufficient energy to keep this place warm, in addition to providing power to run lights, computers, etc. You'll need to grow food without sunlight, but that shouldn't be tough: you need artificial lighting that reproduces sunlight well enough to "fool" the crops. While it might be expensive, I don't think any of that is beyond present technology. How many people you could sustain this way would depend primarily on how much time you had to prepare, I think.
I don't see anything fundamentally impossible about it, so if there are points where we don't know how to make this or that work, the "with sufficiently advanced technology" should solve that.
Presumably without energy coming in from the Sun such a colony could not survive forever. But presumably the Sun cannot last forever either. As I say, I haven't done the calculations, but I'd guess there is enough energy in known fuel sources in the Earth -- coal, oil, nuclear, etc -- to last for, well, I guess it depends on how many people we are supposing are surviving this way, but even if it was millions, there's enough energy to last for centuries if not millennia. Plenty of time to develop the technology to mine the asteroids or travel to another solar system. Assuming all the stars in the universe haven't gone dark in your scenario.
If this happened tomorrow, I think the only place people might survive without energy from the sun is Iceland. The population is small, they already generate a significant proportion of their electricity from geothermal power, and they are used to cold weather.
I'm not considering oil rich places like Alaska because that is really stored energy from photosynthesis. But if we do consider them, the South Pole research station is another place that might hold out, as there are a lot of smart people there, though maybe not with the right skills.
The other thing is, keeping warm in a geothermal area is significantly simpler than in an oil-rich area. All you have to do is dig down! The Maori of New Zealand cook food underground using rocks heated by fire, but the practice surely originates from geothermal cooking.
In a matter of days, though, it's going to get really cold, and carry on till it gets to about the night time temperature of the moon (approx -233C.) That's below the freezing point of nitrogen and oxygen. Obviously it could take decades for the oceans to freeze, so I would expect the temperature to hold at a plateau somewhat below zero C for a very long time
So the people of Iceland are going to have to be very resourceful and build an insulated biosphere in record time, and then will have a few decades to make it airtight.
Food is likely to last from months to years, but ultimately it will have to be grown inside the biosphere using geothermal electricity.
I think they could possibly survive, but it wouldn't be much fun.
Given more advanced technology (fusion reactors?) and a decent warning, I think humanity could make a good go of it, but living on a sunless Earth would be considerably more difficult than living on present-day Mars.
