There's something about octopuses that captivates people, so alien to us and yet also so familiar. Despite a billion years of evolution separating us, they can be strikingly similar, showing curiousity, playfulness, intelligence. It's telling, in an anecdotal way, that aquarium workers often give names only to sea mammals and octopuses. In fact, many scientists who study them believe that they even have individual personalities and can recognize different people. Capable of using tools, solving problems, and remembering solutions, it's easy to see why the idea of an octopus (or more generally cephalopod) civilization is so fascinating.
But would such a civilization be possible? Cephalopods have been around for hundreds of millions of years--far longer than mammals, let alone our own species. Which begs the question: if a cephalopod civilization were possible, where is it? Unfortunately for all us nerds who love the idea of a intelligent tree octopus, there are many major hurdles to the development of a technological octopus civilization. While they certainly have notable advantages--dexterity, intelligence, and mobility--they also have several disadvantages.
The first major impediment to the development of a cephalopod civilization is their short lifespan--most live only a handful of years and then die. Even the infamous Giant Squid, which can grow up to 15 feet long, is estimated to live for only 5 years. Additionally, nearly all cephalopods die after spawning, significantly limiting the capability to build up knowledge from generation to generation. That said, there are a few iteroparous cephalopods and a large number of species whose reproductive behavior we know nothing about, so it's possible that some existing or future species could develop a lifestyle which includes significant parental care and teaching.
Even for those species that die after reproducing it would still be possible to pass knowledge through writing, and here cephalopods may have an advantage. All cephalopods (with the exception of the Nautiluses) have the ability to change the color and texture of their skin, enabling a complex body language. For humans to develop a writing system, we had to learn to abstract the verbal sounds of language to a visual form, but cephalopods wouldn't need this extra layer of abstraction. Early cephalopod writing might be represented by a series of colored patterns or shapes, mimicking their natural communication. And like much body laguage (such as the human smile) it is instinctual, making it plausible for some information to be passed from generation to generation without instruction. This may be an adaquate workaround to pass limited knowledge, although more abstract concepts would require each successive generation to decypher the language.
Civilization is, by definition, fairly social. Being able to read the intentions of others, deceive, and navigate social hierarchies all require significant brainpower. This is likely why all of the most intelligent animals are social. Cephalopods are a diverse group, with individual species both enormously gregarious or solitary. Octopuses specifically are almost exclusively solitary, although at least one species has been claimed to live in social groups. Squid are much more frequently social, often forming shoals of thousands of individuals. The Humboldt Squid (which I like to think of as The Velociraptor of The Deep) is a large squid which exhibits curiousity and perhaps even cooperative hunting. On the other hand, it's also infamously cannibalistic, a significant downside to being social.
But perhaps the biggest hurdle for a technologically advanced octopus race is fire, or rather, the ability to harness energy. There are relatively few opportunities underwater to harness the highly-concentrated energy necessary for metallurgy. In air, achieving a fire of several hundred or even thousand degrees is possible without too much effort, but it is much more difficult to reach similar temperatures underwater due to its greater thermal conductivity. And since metalworking is the basis of most advanced technology, whether advanced mechanical or even electrical, it seems unlikely that an aquatic species of any kind would develop this kind of technology.
Thus, for an advanced civilization to emerge, it would likely have to move onto land, or at least be capable of limited labor on land. Which isn't so far fetched, really. Octopuses in tidal zones have been observed crawling significant distances between pools, searching for trapped prey. And even underwater, octopuses frequently travel by 'walking' across the sea floor--a style of locomotion that is believed to have been shared by the lobe finned fishes that were the ancestors of the first amphibians. While most octopuses have completely lost the rigid shell that is a hallmark of molluscs, squid and cuttlefish maintain this minimal skeleton inside their bodies. Evolution being nothing if not efficient (or lazy), the genetic code for these traits likely remains present in octopuses, making it relatively easy for them to reacquire them. Further adaptation of these structures could enable more lengthy stays on the surface--witness the African Giant Snail which lives its entire life out of water.
That is not to say that any of these disadvantages would preclude a socially advanced culture, just a technologically advanced one. The Aztecs famously used obsidian for nearly everything, from tools to weapons, and developed an advanced urban culture without metallurgy. (Metallurgy was known in pre-colombian America and used by many cultures, but the abundance of obsidian in Mesoamerica and the superior sharpness and hardness of obsidian compared to metal tools resulted in it being mostly unused by the Aztecs.). Similarly, Inca civilization lacked writing (although some archaeologists believe that quipu may be a undeciphered form of Inca writing) and no pre-colombian culture made wide use of the wheel. Despite this, they formed large, culturally advanced empires that existed for hundreds of years.
So, octopuses in space? Probably not. But underwater societies filled with curious animals herding lobsters and fish, and tending fields of clams? I, for one, welcome our cephalopod overlords.
Sources and other cool shit:
Humboldt Squid
Cephalopod Reproductive Strategies
Octopus Intelligence
Octopus Tool Use