Normally, technology in numerous fields will advance due to breakthroughs driven by a need for something like more efficient weapons or farming. How can I explain a civilization being "technologically stagnant" and having themselves stuck with technology found around the time ancient Rome existed (600 BC). I have theorized that they can't advance to using electricity due to solar flares constantly bombarding the planet, although I don't know if this is plausible. Also the world they live on is an Earth analog, although it has no fossil fuels present.
You've pretty much answered your own question insofar as if there is no need for technological advancement, there won't be any.
If your farms are producing all the food you need and the climate is consistent and temperate all year around, there are no barren areas, no strategic points of coastlines or ports that are envied by the rulers of opposing nations, no resource shortfalls...
...you get the picture...
Then there's no need to develop anything like better weapons, ploughs, or technology in general. Putting this another way, there is no reason to advance if your life is fine as it is.
This in point of fact leads to an interesting anthropological theory I once heard that said that technological advancement only happens in cold climates. The reason was that the cold made life uncomfortable, and provided a forced scarcity of food over a winter period. This meant that people strived to find ways to make their lives more comfortable and as such, developed and refined new ways of doing things and new tools to do them with.
Regardless of that theory, if you look at the relative technological level of European explorers and African tribes in the 18th and 19th centuries, it is clear that Europeans with their harsh winters and relatively scarce resources had advanced more than the African tribes with their temperate climates and a bountiful and relatively constant food supply.
So; make your civilisation relatively happy and content, and advancement won't be as fast as if they're struggling and uncomfortable.
All ancient civilisations were essentially shaped by theology, so you simply need to make yours prohibit - and severely punish - technological advancement. I suggest you refer to the Safehold series by David Weber for an excellent example of how this could be achieved.
I have a few thoughts on this based on groups on some historical context (and their modern philosophical descendants):
- Groups which shun technology for religious reasons
- Groups which cut themselves off from the outside world due to fear
- Groups which fear technology itself and were it will take us in the future
Amish - religious angle
Most of the Amish have hit a stopping point when it comes to technological advancement (they are not adversarial toward it but severely limit its use). They've chosen to live simply in order to better serve their religion and idea of what it's god would wish. Indeed, this does not only limit technology use but limits the needed education (most stop school at 8th grade) that would be required to engineer new devices/tools.
Isolation or Fear of outside influence & loss of control
Though certain specific technologies grow better with war, new general technological advancement requires periods of peace (see statements by Sir Henry Tizard and Sir Stanier, pgs.7-10, Peter G. Klein's statements, and linked articles/talks).
However, when pursuit of that peace causes such a fear of returning to war that governments start to impose heavy restrictions on its populace and actively force out any outside influence (pg.13) to ensure the power base of their own government - it tends to squash any ideas or technological development due to fear it will lead to revolutionary ideas or someone gaining a powerful "weapon"1 the government does not control. It is basically trading growth for stability - at least until someone starts shooting cannons off your shore.
Fear of technology
There have always been those who prefer to live "off the grid" and those who fear what new technology will bring. One can look at Henry David Thoreau's Walden and the transcendentalist movement of the late 1880s and see elements of these ideas. While the Luddites of that same era - actually smashed new technology out of fear it would eliminate their livelihoods.
Or why not all three
It would not be hard to imagine a group which saw these driver-less cars coming (lets say Uber and Taxi drivers) starting a movement against this specific technology, began excluding countries and peoples who supported them. Then being expanded their philosophy to slowly include all technology as evil ("un-natural") and eventually take on religious undertones as justification for their fears.
1: Weapon here could be an actual weapon but also any form of new technology which allows you to generate income, food, or even good will at a rate that allows you to be a threat to those in power (whether you intend to us it or not)
None of what they have is their own technology. They barely understand what they have.
"We look for things that make us go."
One of my favorite Trek TNG episodes. The Enterprise encounters a broken ship. Geordi goes over to help and the problems he fixes are very simple. It turns out that the aliens are very, very slow-witted. They understand none of their tech because it is not really theirs. They acquired it from a different race.
So too your people. They inherited their tech and they greatly appreciate it but do not understand it and maintain it only through careful rituals. Possibly they were refugees and found the tech on the planet they came to. Possibly their ancestors were more intelligent and creative than they are, but like the Eloi in HG Wells The Time Machine, the descendants can use and to some extent maintain the tech but cannot invent anything new.
Progress happens wherever there's room and necessity for experimentation - e.g. where you won't starve to death if your experimental agricultural technique doesn't work as advertised, but as the competition is high, you get a lot of incentive for trying something new.
You can place your civilization in a particularly harsh environment like the Extreme North. Make them originate as settlers from an early Iron Age civilization. Place them far from the sea so that trade isn't an option, and plunderable neighbors are hard to reach. Make the setup very stable (so that disasters don't happen on a regular basis) and isolated - say, these settlers had to cross a stretch of desert or tundra that became impassable centuries later due to climate change.
Now, your civilization struggles to maintain its current technology but still has to preserve it. Innovation is perceived as unnecessary and outright dangerous. Any change in the ways of life can either starve a commune to death or threaten the authority of the local government. This is a perfect deadlock that has been observed many times here on Earth.
Alternatively, since this world isn't Earth, you can make your civilization consist of regressed colonists - i.e. the remnants of a colony that went through a catastrophe (ranging from a long volcanic winter to a plague to a simple crop failure) and lost most of its technology over several generations. Then the technological stagnation might be explained by the fact that they simply have to recover their numbers. If they manage not to forget writing and agriculture, and avoid genetic bottleneck effects, you could expect them to rise again within several centuries - but until that time they would appear to be stagnant.
There are many examples of technological stagnation through out history.
Many uncontacted aboriginal groups in Brazilian forest have never advanced into the iron age. They remained in a primitive state for various reasons.
Some of the most advanced native American cultures never advanced, in many ways, much past the early bronze age.
When the Tokogawa Shogunate came to power, they closed their doors to the world. When they reemerged 250 years later, they found a world that advance well beyond them, while they made very little advancement.
Pretty simple solution. Deny access to higher education. Or make it only for an elite group, several countries doing this right now. People can only work with what they actually know or have access to learning.
No written language.
Most of the technology of 600 BCE could be passed orally from one generation to the next with little room for advancement over time, because even if you did invent a one-off advancement, it would take a lot of work to explain to every other person you want to teach it too; so, it often just dies with you or it takes many many generations to become commonplace. Moreover, if you invent a single piece of a bigger more important puzzle, you don't know what else is out there to build on to complete the bigger puzzle in your own lifespan; so, it disappears with you because no one knew how important it would one day be.
Much like cavemen existed for 100s of millenia with just basic tools and weapons, if cavemen discovered smelting and farming before writing, then the bronze age would have been just as static.
In this case, you don't need to add artificial ceilings through culture or nature, you just hit a ceiling that you can't break without first inviting this one specific thing that no one has thought of yet.
Progress is not inevitable!
It only seems that way because our modern civilization has created a positive feedback loop that continually increases the pace of technological progress. Prior to the various revolutions within the past few centuries, there are plenty of examples of individual civilizations whose technological progress stagnated or even regressed because one or more of the factors import for innovation was lacking.
Key Factors for Innovation
The following is a list of factors that can contribute to innovation (some of which are discussed further in other answers). Understanding these different factors will allow you to craft a scenario that best suits the world you are building.
Put simply, to develop technology effectively civilizations need extra food and other resources. Innovations will be sporadic and difficult to retain if you do not have individuals who can dedicate at least a portion of their time to learning and experimenting.
Necessity may be the mother of invention, but scarcity is often the bane of innovation. When resources are limited, a civilization will need to use them towards survival. Thus, in dire circumstances, it's exceedingly difficult to develop technology to overcome the situation; instead, it's more likely that people will adopt different practices using similar (if not inferior) technologies.
Motivations & Incentives
Though some individuals are natural innovators, the majority of people need some form of motivation. Competition, be it nations warring against each other or individuals participating in a free market, is a great motivator, but it is not the only one. People may also innovate for the sake of prestige, to promote humanitarian efforts, or enable creative expression. Too few people will develop technology if there are no incentives to do so.
To ensure technological progress has a meaningful impact, a society must be open to innovation. If the majority of people are hostile, fearful, or even just apathetic towards new technologies, they will not be adopted. Furthermore, those who do try to expand the civilization's intellectual and technological capabilities will, at best, be ignored and, at worst, persecuted.
Establishing networks of communication is vital to innovation for numerous reasons. Communication can expose individuals to diverse, unique, and inspiring ideas that they were unlikely to come up with themselves. It enables collaboration so that innovators can build off of each other and coordinating efforts (as opposed to working in isolation on the same technologies). When two or more civilizations communicate with each other, it expands the pool of possible innovators and provides a means of retaining knowledge should one civilization fall.
For much of history, this was accomplished through friendly trade and diplomatic exchanges. War may be a motivator for innovation, but peace is an enabler.
Many technologies cannot be developed without access to the appropriate resources. Animal husbandry would be impossible without animals that are suitable for domestication. Metallurgy is completely impossible without metal ores and nuclear physics would be nearly as impossible without sufficient quantities of radioactive materials. These resource dependent technologies are often the most vital to expanding a civilization's capabilities; it's difficult to imagine a modern world without such resources.
Literacy & Education
The value of literacy and education is two-fold: first, it significantly increases a civilization's ability to retain knowledge. Secondly, it enables more people to be innovators. The more widespread these are, the better!
The following are just a few examples to illustrate the above points.
Lack of Surplus, Lack of Motivations & Incentives
It's worth mentioning that the first civilizations to emerge were agrarian societies that settled along fertile river basins. These were by far the easiest places to produce surplus with limited technology.
Civilizations that lived in other environments would not be able to generate the same level of surplus. Meanwhile, non-agrarian civilizations lack incentives for innovation. If resources are scarce or your neighbors are hostile, you can readily move.
Lack of Cultural Acceptance
As for stagnation, one example (as mentioned in another answer) is China. It had the potential to initiate the Industrial and Scientific revolutions. However, the government actively discouraged innovation as it was seen as a threat to the stability of the nation.
Similarly, Arab scholars were not only responsible for retaining much of the knowledge past civilizations (including the famous Greek philosophers that Europeans would later celebrate), but they made a number of important advancements. Unfortunately, the sentiment that scientific pursuits were contrary to the teachings of Islam grow to the point that the various Muslim academies were shutdown.
Lack of Communication Networks, Lack of Natural Resources
In the Americas, civilizations like the Aztecs and Incas emerged. They were reasonably complex, but geography and lack of natural resources greatly undermined technological development.
The Aztecs and Incas were contemporaries, but they were far too removed from one another to effectively establish trade. Furthermore, with the exception of llamas, there were no large mammals suitable for domestication; they couldn't benefit from increased agricultural productivity provided by various beasts of burden nor the nutrition provided by other livestock found commonly throughout Eurasia (and Africa to a lesser extent).
Lack of Literacy & Education
To an extent, every civilization prior to modern times could serve as an example. The implementation of compulsory, universal education opened the door for so many more people to become innovators than before; without it, scientific and technological development would remain the hobby of elites and not the purvey of dedicated professionals.
Given all of the above, what might cause your fictional civilization to stagnate?
Scenario #1: A Fat, Lazy, and Isolated Society
Imagine a map of Rome at the height of it's power. Now imagine that, instead of more land beyond the edge of that map, there was nothing but ocean.
On this fictional continent, there were many difficult cultures at first. With the advent of sailing, though, the world started getting smaller. Half a dozen rival civilizations formed, competing with each other. This lead to considerable technological development.
Eventually, though, one civilization became dominant. It conquered the rest, gaining control of the most productive lands throughout the continent. Competent rulers ensured the stability of the empire, expanding it to the point that its territory covered all but a small fraction of the known world. The peace and prosperity ensured the gradual assimilation of other cultures; without any significant outside influences, the empire became rather homogeneous.
With everyone content with the status quo, change became regarded with incredible fear. Why risk it when so many could enjoy an indulgent, hedonistic life style? Everyone's efforts became guided towards preservation, sustenance, and pleasure.
Scenario #2: Ecological Decline
In this scenario, the stagnation is due to a change in the environment, such as the onset of an Ice Age or a shift in the prevailing winds caused by rising mountains that reduce precipitation. The decline is too subtle for most people to notice, so little effort is made to combat the change. However, the population gradually declines as lands become less productive and domesticated animals struggle to adapt. Your civilization is experience an agonizingly slow death that will take millennia to realize.
Since this ecological disaster affects the whole known world, your civilization's rivals are unable to exploit the situation. Occasional attempts, however, may be enough to encourage your civilization to maintain what technology it does have, though.
If your civilization is not prosperous nor inclined to change, it will stagnate technologically.
Limiting Information During the dark ages, most books were not accessible to the public. Education was limited, common folks are mostly if not illiterate. Even new discoveries at that time was labelled as witchcraft, causing people to fear discovering new technology. This was because the church had power over the people.
If you limit the distribution of information it will be easier to control the mass and limiting their potentials to grow.
Dr. Slump has a perfect example of this: Gatchan, a self-replicating Angel that eats metal.
Gatchan is an Angel born from an egg placed on Earth by the Kami of the galaxy during the prehistoric ages, to prevent further development of the human civilization seeing that other civilizations eventually destroyed themselves and the planets they lived on. Gatchan's ability to replicate itself as well as its fondness of eating metal should have ensured that humanity would remain primitive and innocent.
However, in this particular story, the protagonists happen to come across the egg during a time travel trip to prehistoric times, and end up taking it back to the present, thus subverting Gatchan's original purpose. (Some causal time loop hijinks here.) Kami is considerably confused when he arrives later on to check on Earth and discovers that it had become technologically advanced after all.
Lack of Competition.
One of the biggest incentives to come up with new technologies and new ideas is if you are competing with your neighbors for resources. Europe, with more than a dozen different power bases in a relatively confined area, was constantly dealing with rival nations looking for some kind of an edge. Once you come up with something that gives you an advantage, your neighbors will be forced to do the same to stay competitive. With only so many resources to go around, the most innovative nation is more likely to end up on top.
Compare this to China's situation. Even though they invented gunpowder, moveable-type printing, the compass, as well as many other innovations, all which were instrumental in aiding other nations to make huge technological leaps, they were never used to their potential in Asia until much later. Ancient China had a vested interest in suppressing innovation, as new ideas might give other nations or internal factions enough leverage to threaten the status quo. Hence the reason why in China gunpowder was traditionally used for fireworks, little more than a curiosity, as opposed to cannons and muskets.
This eventually came to haunt them when England and other nations came calling, as their lack of innovation in technology left them vulnerable to more advanced invaders. Japan also experienced this phenomena, for similar reasons.
So if you have one single nation, and no competitors, you have a good reason to avoid technological change.
Resource Constraints (scarcity or cultural) -
Electricity, for example, requires medium to conduct the electricity from the point of power generation to the point of consumption. Copper is the preferred material, due to ease of refining, maleability, etc.
Without copper wires, it's far more difficult to get power from your energy source to your endpoint. It also makes it very difficult to make electric motors.
Make copper something associated with a God or Demon, either way, people won't use it, or make large (industrial quantities) inaccessible.
You do need to be careful with the question worded the way you did. Rome had all of the technologies needed to kick off the industrial revolution (hydraulics via water, understanding of steam, advanced\precision mechanical engineering, understanding of vast infrastructure projects, literacy), they just hadn't put the pieces together quite the right way yet.