Humans with their technology developed on Earth could live on a tidally locked planet where neither day nor night exists.
We used to think that such planets become so-called eyeball worlds, with scorching temperatures on the side facing the star and frigid conditions on the dark side. However, computer modelling revealed that this view was inaccurate and eyeball worlds might be rare. It’s enough that a planet has about 10% as much of certain types of atmosphere as Earth to distribute the heat from the hot side to the cold side somewhat. Conditions would not be as extreme as previously believed, the denser the gaseous envelope is.
Planets facing the same side to the star might be common in the universe. About three-quarters of stars are red dwarfs, which have a tight habitable zone. There is doubt about the possibility of habitable worlds existing in orbits so close. Such stars flare and can damage atmospheres and oceans, but even more massive stars between the size of red dwarfs and the Sun are likely to often host worlds facing their star on the same side in their habitable zones.
We are not sure if the evolution of complex life on such worlds is possible. There might be a need for a different type of photosynthesis. The one on Earth needs some rest time from light. Maybe it wouldn’t be a big problem, which means that if we ever find life on other planets, it might be on a world that constantly faces its star on the same side, since these planets might be common in the universe.
If we ever establish a colony on such a world, it would be advisable to recreate day and night cycles in our outposts. It might be detrimental to our health to experience constant daylight. It could affect circadian rhythms. Medical science is only becoming increasingly aware of the impact on health that it causes.
Not to answer your question with a question… but what is your supposed conditions if it’s neither day nor night? We call the time of the day when the sun is out “day” and the time of the day when the sun has set and it is dark “night.” Are you talking about that time of day when the sun has set behind the horizon but, due to refraction, there is still some light in the sky, for example, as we would see here on Earth during Civil or Nautical Twilight?
The answer to this question will vary greatly depending on the size and output of the star around which the planet is tidally locked, the location of the planet within the star’s “Goldilocks Zone”, and how well that planet’s atmosphere can spread and dissipate the heat.
The side of a planet tidally locked to its star will always be facing its sun, so it will get pounded constantly with the sun’s rays. The opposite side will never face the sun, so it will never warm. Objects in our solar system that have one side always facing the sun have the sun facing side exceedingly hot (to hot to live in) and the side always in darkness get bitterly cold (to cold to live in). So you would have to either be in just the right spot that the side always facing the sun gets just enough energy to provide sufficient warmth for humans to survive, without trapping too much heat so the temperature continues to rise. Or, you could have a sun facing side that is way too hot, but an atmosphere that is able to spread that heat over to the dark side so its warm enough for humans to survive. But that does not answer your question, because one of those sides would be day, the other night.
The best chance would be on a planet where you find a temperate zone right on the terminator line between night time and day time. The sun facing size would be impossibly hot and the dark side would be freezing cold, but right along this thin strip along the terminator where day turns to night, it might be possible to have a temperate zone where humans could live. Even though we would probably still consider it “day” since the sun was still lighting the sky, I guess you could qualify that and say it’s actually that period in between day and night, or twilight.
