First of all, there's the year that the glowing baby was born. There's no way of knowing what year it was, but since technology isn't that much more advanced than ours, and we're told it was halted by the appearance of quirks, we can safely assume that it was in the early-to-mid 21st century. Then there's the fact that we see snow in Japan, a near-equator country getting hit by warmer winds from the ocean. Since dealing with quirks tore the world's focus away from other issues such as climate change, I think we should definitely assume that the world had time to deal with humanity's massive carbon footprint, something that, given our current situation, would likely take twenty or so years to properly deal with. Yes, I know there's the horribly trashed beach that All Might had Izuku clean up, but litter and greenhouse gases are two different things, and given the current real-life situation, I think it's safe to say which one the world prioritized. Besides, a lot of that could easily have been floating in the ocean for decades. All in all, I think the glowing baby was born around 2050.
Then there's the initial backlash and the appearance of All for One. It likely took the world between five and ten years to fully react to the initial appearance of quirks, and All for One appeared pretty early on, so we'll say that All for One appeared about ten years after the glowing baby, around 2060.
Now for how long the quirk society has been around. The doctor specifically states that Izuku is a part of the fourth generation. In the developed world, parents tend to have kids near middle age, so I think it can be safely assumed that each generation is around thirty to forty years (we'll round it at thirty-five on average). If Izuku was born early in his generation, we can probably say that he was born around 110 years after the fiasco with All for One and his brother, putting us at around 2170 when he was born. This checks out with All for One having been in power for over a hundred years.
Now, one reason you may think it doesn't check out is the fact that Izuku is the ninth holder of One for All, something we previously said appeared around 2060. There are those who draw a pattern from the fact that Toshinori and Izuku both received One for All at the age of fifteen, and saying this probably went for everyone. However, they are special cases due to them both being quirkless kids in a superpowered society that wanted to go to a high school for heroes - something that kind of required a quirk. Most of the bearers were likely adults when they received it, due to experience as a hero and with the quirk they already had being a good mix with something so powerful. They likely received it at a later age, shortening the time they had to use it. Most notably, All for One's brother probably didn't hold on to it for very long, as he knew early on that he would need to pass the quirk on to future generations. With All Might having probably had the quirk for around fifteen years before Izuku was born, and the first user probably holding it on for ten to fifteen years, this puts the average time of holding the quirk at fifteen years for the others, though some may have passed it on prematurely due to injuries or expecting to do something that would get them killed, so I think it's safe to assume that, in this timeframe, the median retention time of One for All was around twenty or so years.
So with all of that, in addition to Izuku being fourteen at the beginning, fifteen at the current point in the anime, and sixteen at the current point in the manga, I believe that My Hero Academia takes place right around the turn of the 23rd century.
