This depends somewhat on the nature of the factual inaccuracy and how easy it is to verify the truth of the matter.
If the inaccuracy is provable by evidence found in what Wikipedians consider to be reliable sources, then you can just remove or correct the claim, citing the appropriate source, and if there is a dispute it gets discussed on the talk page or, if consensus is not reached there, at some other discussion page where there are more eyeballs.
If the inaccurate version is the mainstream story as found in the bulk of recent reliable sources (like scholarly books and newspaper stories), then correcting it is harder: you need to find sources that are even better than those, e.g. by a respected historian who specializes in something topical writing a peer-reviewed journal paper. It works even better if you wait for a scholarly consensus to crystallize around the improved version of the story. In the mean time, the language of the article can usually be changed to present something neutral to readers, e.g. saying that the traditional story as presented by scholars X and Y is A, but that scholar Z disputes this and proposes B instead.
But then what's the actual process to get a factual inaccuracy corrected?
This depends somewhat on the nature of the factual inaccuracy and how easy it is to verify the truth of the matter.
If the inaccuracy is provable by evidence found in what Wikipedians consider to be reliable sources, then you can just remove or correct the claim, citing the appropriate source, and if there is a dispute it gets discussed on the talk page or, if consensus is not reached there, at some other discussion page where there are more eyeballs.
If the inaccurate version is the mainstream story as found in the bulk of recent reliable sources (like scholarly books and newspaper stories), then correcting it is harder: you need to find sources that are even better than those, e.g. by a respected historian who specializes in something topical writing a peer-reviewed journal paper. It works even better if you wait for a scholarly consensus to crystallize around the improved version of the story. In the mean time, the language of the article can usually be changed to present something neutral to readers, e.g. saying that the traditional story as presented by scholars X and Y is A, but that scholar Z disputes this and proposes B instead.
Publish a blog post or a tweet. Then you can cite that.
Wouldn't your blog post or tweet be considered a non-notable or unreliable source?