A question that has occupied many a lore fan.
The "TES I: Arena" player can find the location west of Whiterun. It has an official map marker and can also be found on created by fans.
The location Blackmoor is also present on a map.
Strangely enough, the location at this exact spot in "TES V: Skyrim" is called Fort Greymoor. Just like on the old maps, this fortress ruin lies west of Whiterun, making it highly probable that it is the same location.
If you also factor in the biography of Barenziah, it becomes completely confusing. There, a location not far from Whiterun is described, which is called Darkmoor. It was the seat of Count Sven Advensen and Countess Inga, who looked after the young Barenziah as her guardians. Barenziah's escape from there is described as follows:
Quote from "The Real Barenziah":
"One fine spring night she and Straw took two of the finest horses from the stable and rode through the night to Whiterun, the nearest city of any size, and where Straw wanted to be. But Mournhold and Morrowind lay to the east too, and drew Barenziah like a lodestone draws iron."
The official (Imperial) biography describes it the same way:
Quote from "Biography of Barenziah":
"That very night she disguised herself as a boy and the two fled to the nearby city of Whiterun."
Therefore, the fortress Darkmoor is likely the fortress Fort Greymoor known from the Skyrim game. And if that is the case, then based on its geographical location, it is likely the location Blackmoor. But why these confusing name changes?
As is well known, the geography of Skyrim has been altered here and there by Bethesda since Arena. One could therefore argue that the original Blackmoor was located somewhere else entirely (such as High Hrothgar or Markarth) and bears a completely different name today. However, if one looks at the first version of "Biography of Queen Barenziah" in "TES II: Daggerfall", Barenziah spent her childhood with Count Sven in
Blackmoor. It was not until the edition in "TES III: Morrowind" that the location was named
Darkmoor. One can only guess at the reasons for the name change, but this could be the definitive proof that Blackmoor equals Darkmoor.
Note from Almanac moderator Deepfighter:
"Many of the DF books were rewritten for Morrowind. Especially the lore-heavy books. The editor probably liked Darkmoor better than Black Moor, which I can understand. The fact that this is the same location is indisputable to me.[...]"
What about the newly introduced
Greymoor in "TES V: Skyrim"? The fortress is located exactly where one would expect it based on the description in the biography. But there is another, hidden clue. There is a "Fort Greymoor Prison Key". Although it does not exist in the game, it was likely intended for something during development. In the original files, it is named "Fort Blackmoor Prison Key" (00016FF1, CWFortBlackmoorPrisonKey). This should be the final missing link between Blackmoor, Darkmoor, and Greymoor, and proof that it has always been the same location.
The area around Whiterun has not yet been in Elder Scrolls Online (as of January 2021, Update 28), so it is not yet known what the location is called there.
However, the location discussed here must not be confused with the already existing in ESO, which is located far away and in Blackreach.