In My Hero Academia, few characters are as beloved as Shoto Aizawa. At first appearing as a very strict teacher and almost unlikable character, by the end of the first season, he had won over most fans of the series. One aspect of his character that almost immediately endeared him to fans was his perpetual desire to sleep, which many of them related to.
Going by the hero moniker "Eraserhead," Aizawa spends most of his days teaching the students of class 1-A during the day and works as a pro-hero during the night. That means, there is very little time in his schedule for sleep. But considering that the rest of the teachers at UA don't seem to have this problem, why is it then that Aizawa does.
Since the very first season, Aizawa has been shown to be quite the capable combatant as well as the kind of hero who took the job to help people, not because of potential fame and riches. In fact, Aizawa's frequent disregard and hatred of the press practically ensures that he could never be that type of hero, even if he desired to be. This means that he's the type of hero who will fight whenever he is needed, much like how All Might had been before he lost One For All. That being the case, it's easy to see Eraserhead pushing himself past his limits in order to help people, which could be part of why he's always sleepy.
This is partly due to the death of his dear friend, Shirakumo Oboro. After his friend's death, Aizawa was left traumatized. So afraid that he would lose another friend in a similar manner, he dedicated himself to his studies and training. Performing his duties as a student in the day, he would stay up throughout the night training his body, hoping that it would be enough to keep the people he cared about safe. It's very likely that his routine continued even after he graduated, with training being replaced with hero work. It's even possible that he just added it onto his routine instead, making it very easy to overextend himself.
On the subject of spreading himself too thin, he is also responsible for Eri and Shinso's private training. With Eri, he is responsible for making sure her powers don't go out of control again, and while this would only take a glance on his part, it still requires his time and attention which can be draining. Shinso is a whole other matter, being that Aizawa is training his physical abilities. While not an actively dangerous combat situation, the class 1-A teacher still has to exert energy in order to show Shinso how to fight or use the binding scarf.
There is also the matter of his quirk, or rather its shortcomings, that probably plays a part in his tiredness. Setting aside how quirks are known to drain stamina, Aizawa's quirk isn't as overpowered as it may initially seem. The ability to turn off an individual's quirk just by looking at them might come across as invincible, its weakness is in its simplicity. Even though he turns off their quirk, the foes aren't necessarily incapacitated. The stare will only disable their powers, not subdue them, meaning that if the villains are able to move and escape or have naturally strong bodies, his quirk won't be of much use.
Eraserhead doesn't have any other kind of physical enhancement from his quirk the way heroes like Endeavor do (while his quirk his is fire-based, Endeavor has shown on multiple occasions to be superhumanly strong and durable too). This means that his body is only as strong as a normal human's who engages in the same fitness regimen as he does. Because of this and the nature of his quirk, his body will be experiencing fatigue much more frequently than others, which would explain his overly exhausted demeanor.
Aizawa is a true hero through and through. Spending his days training the next generation and nights protecting the innocent. While he doesn't always come off as the friendliest guy in the world, he is easily one of the most dedicated heroes Japan has. After sacrificing as much as he has to keep his friends, students, and the people of Japan safe, Aizawa has definitely earned the right to a nap.