What if our planet was always lit so that there would be no nighttime? Assume that the scenario is like this, where A is the planet and B and ABb are the stars. And they are all aligned so that it will fit the scenario.
This planet has stars on both sides that gives enough light to keep it warm yet not enough to burn the surface. Will it still be necessary for us humans to sleep regularly like we do every night?
The solution is to have the animal evolve to sleep like a dolphin, with half their brain at a time, though for different reasons. Dolphins on Earth do this to avoid drowning, but the concept is still the same. Without a method like this, your people will have a hard time living without sleep related issue, like the dolphins said "So long and thanks for all the fish, so sad that it must come to this."
This opens the door to some interesting psychological and physiological features;
When you get started here, you can really get rolling. I can imagine a species that instead of sleeping half a brain at a time; lives on quarters, or even eighths at a time. Keep in mind that if they evolved in this world, they wouldn't even have a concept of night; never mind associate it with sleep. Of course, if you have your world suddenly have no night, then this answer is pointless.
A very brief overview of something I remember from my Psych class.
The reason for sleep is still somewhat debated and unknown. Three of the leading theories are Evolution, Restoration, and Processing. The evolutionary theory states that animals evolved to sleep during periods that it would be dangerous to be awake (ie. The time most predators are awake.). Restoration is exactly like it sounds like and is the theory that we need rest in order to repair the body and mind. Processing is that we need sleep in order to process everything we did during the time we were awake and prepares us for the next awake period.
Because of these theories, it would be doubtful that humans would not need sleep if there was no night. If you wish to understand sleep more thoroughly than my brief overview has provided, I would suggest looking up sleep theories, theories of sleep, or a psychology book as that talks about sleep too.
Hope this helps :)
Edit: I would also like to add that sleep seems to play an important role in mental and/or physical development. We sleep a long time as babies when our minds and bodies are still developing and as we age, the time we spend sleeping or needing sleep decreases.
You need to look up the classic story Nightfall by Isaac Asimov! Furthermore, look into the backstory of how it was written, which is essentially your question.
People like us evolve to live in constant light. The only twist is that night does fall once in a thousand years and the stars come out. This drives the people crazy.
They do sleep, as he wanted to make them just like us for the story. That is your choice, so the question has no real answer. You can have your characters sleep or do something distinctly different that serves a similar purpose. If so, you can get creative about what kind of cycle to follow. You can have them never sleep at all. You can do something like migratory birds do and have them rest half the brain at a time while remaining active.
Perhaps different species will do different things, so you can explore multiple solutions in the same story.
There are areas on Earth which regularly don't get any nighttime for part of the year: the polar areas. While this effect is not enough for evolutionary timescales, as in to balance things out, it's only nighttime during the other season, people do sleep at night even though the sun is up. Some people do use curtains, blindfolds etc. but you can actually get used to sleep while it's bright outside.
Nobody fully understands why we sleep. There's plenty of pieces that we've put together about what sleep does for us, but we don't fully understand the puzzle. We also only have observable examples of sleep from 1 planet, which happens to be dark for half of the day, so we don't really have any good ways of guessing how things would have evolved without a day/night cycle. Given the "nature finds a way" mindset, one might assume that life on such a planet would figure out something. There's plenty of really odd sleep approaches out there. Crocodiles, for instance, can sleep with one half of their brain at a time so that they don't get surprised.
However, if we're talking about humans, as in you transplant some of us to a planet with an eternal daylight, the answer is easier. Humans shut down without sleep. The structures coded for in our genome assume a day/night cycle, and our brain simply doesn't work without it.
While I'd rather ponder the possibilities of a planet suspended in the Lagrangian point between the stars and mechanisms that lead to such configuration, here's what I think about sleep.
It is not necessary for humans to sleep every night, 8 hours straight, or even regularly(though regularity is beneficial). While sleep deprivation does affect people very much like narcotic substances (diminished motor control, grogginess, up to even hallucinations), the need to sleep does not come from the changes in light levels. I've personally experimented and realized that about 28-hour cycle suits me better than the 24-hour locked-to-day cycle. During the time I didn't see the sky to assess the time of day for long stretches of time, so my body forgot the regular cycle and fell into what was "required", rather than timed by the setting and rising of the sun.
There are several different sleeping patterns you can employ to alter your cycles, some claiming to be more efficient(requiring less sleep per 24hr cycle) than the solid "8 hours at a time" we've grown to hold as the status quo. Here's a blog of a person switching to 20 minute naps for every 4 hours, one of the most extreme polyphasic sleep cycles that maximizes the time you are awake per day. As he explains it too, while you might get more active hours to your day with a "powernapping" sleep schedule, the downfall is that those smaller windows of sleep need to be extremely efficient. After about two weeks of extreme sleep deprivation(not reaching REM sleep during the naps) his body adapted and he'd be able to get to REM stages within minutes of falling asleep. The final post(A year after he ended the experiment) details some of these pros and cons.
Should you have no clear cycle of day and night, then time would simply not measured by the changes of light levels, but perhaps by some contraption that had a really reliable interval in it's action. This would also perhaps create a society where people are awake at whatever hours suite them, rather than the uniformed cycles most people on earth follow. A polyphasic sleep cycle might easily be the norm in such societies, since there would be no clear distinction between times of "day".
I was just brainstorming this on your behalf. Note, these are options other than binary star system. Just things you can consider.
We'd probably sleep like we do more and more nowadays - taking a nap throughout the day as and when we feel sleepy. This is quite normal now, its called polyphasic sleep and many animals sleep in this way, 'napping' during the day and night.
It also happens with humans too, several countries where it is generally very hot in the midday, have a siesta, where they sleep for a while. This biphasic sleep pattern is also normal - before electricity, TV and internet distractions we all used to sleep in 2 phases, often in the late evening and then after waking for a hour or so, again in the early hours. Notably mentioned by Pepys as "little sleeps".
There are other, more unusual sleep patterns
So I think all that would happen is that sleep patterns would change to be more spread out, more chunks of sleep spread out through the long day. 2 hours after lunchtime and again after tea in front of the TV (no change there then!) or irregular sleep patterns as and when the opportunity arises.
Here is one example system with an Earth-like planet and 5 Suns (from https://planetplanet.net/2016/03/22/an-earth-with-five-suns-in-the-sky/):
In this system there is an 8 year stretch in which there is no night time on the planet. (For details see here: https://planetplanet.net/2016/03/23/earth-with-five-suns-in-the-sky-when-would-night-fall/)
This is clearly a special kind of system, but this kind of thinking could easily be expanded upon to create a system in which there is no night time at all. In fact, if this is of interest to anyone please let me know.
What would the sleep cycle of a species living on an all-night planet? Would they sleep all the time? I don't think so.
The "cycle" might look less cyclic as every species has now the freedom to develop its own rythme and adapt it to its own needs, but also in accordance with other species (pries and predators). So it can get really messy and complicated, compared to our relatively orderly system.
That said, it is possible that life evolves without ever 'discovering' sleep, creating other restoration systems which are more efficient and working in background mode for 24 hours (or whatever day they have).
Another way to look at it: people who live in places where it gets 6 months of night and 6 months of daylight. Over the centuries those peoples were living there, we would expect them to develop a slightly adapted life cycle. But as far as I know, they didn't.
