Interesting question. This depends on the time/era, as the city itself has fluctuated quite a bit with wars, invasions, and events. We also have several canon sources and hints as to the city's size, but a lot of these contradict eachother. The Imperials love their census, but sadly, I do not believe we ever truly get a solid number. Also, the depictions of the city vary throughout the games, as do many repeated locations.
194 people live in the city in Oblivion, along with 119 guards as well, bringing the population up to at least 313. I'm not sure if generic NPCs and traveling NPCs are counted. This is clearly a much larger population than the housing we see in-game, and a lot of NPCs just kind of stand around and never go to a "home", so it's unclear if they actually live there. However, this was obviously all done due to hardware limitations, and the sheer scope of having a realistic size of the city and a population on the screen. I would not use this as the canon number, as every other game describes the Capitol as being huge, and in Oblivion it's barely bigger than most Morrowind or Skyrim towns. We can easily say this is the bare minimum of the population of the Imperial City, however.
I'm not sure how many live there in Arena, and honestly it would take a long time to play to that point and find out. Still, the Palace itself looks way different in Arena, appearing much larger than it does in Oblivion.
I know nothing about ESO, except that it takes place in the past before any of the games, so that information might not even be that relevant. MMO cities are usually misrepresented in terms of population anyways. If someone could chime in on this, that'd be great.
Depending on which size comparison you use, like this one or this one by Kirkbride, it would be easier or harder to fit a lot of people onto the island itself. It's supposed to be smaller than Vivec, and bigger than Mournhold, just based on maps of those locations and their size in the games. I'd say the population is between those two cities, especially considering how vertical and compacted the housing in Vivec is compared to the Capitol.
The Elder Scrolls Wiki just says "several thousand", without any citations or sources, so I'd dismiss that number. Also, that could mean anywhere from like 3000 to 999,999, which is obviously a big span.
Opinion here: Personally, in my head-canon, I'd say something like 80k people based on what I've read, how people talk about the city, and how much population it would take to keep the Empire's capital going and defended. The cities and towns in Tamriel are sparsely populated by any standard, and while this is likely mostly due to engine limitations throughout the games, I still think that's just how the universe is - sparsely populated.
In conclusion, unless there's a book, letter, or dialogue somewhere in a game that directly says so, all we have to go off of in-canon is Oblivion, Arena, and some estimates. I don't think we'll ever know unless we revisit Cyrodil.
