There are two issues here, one is ensuring that the erratum is still online somewhere many decades from now, the other is ensuring that it can still be found. The first issue is actually simpler to resolve than the second.
Concerning the first issue, the persistence of a link, the InternetArchive (a.k.a. the "Wayback Machine") will make a snapshot of your file and store it indefinitely. They crawl the internet and may eventually find your file, but you can submit the URL to make sure it happens right away. Wikipedia makes extensive use of the InternetArchive to restore access to broken links.
The second issue of findability is more tricky. The study Linking of Errata: Current Practices in Online Physical Sciences Journals showed that many publishers do not do a good job of ensuring that the reader of the original article is directed to an erratum. They recommend the development of standards for the linking of original articles to errata, but these have not yet been widely implemented.
These two issues apply to any field of science, but mathematics is somewhat special because of the "eternal" value of a proven theorem. The presentation "Towards a world digital library" makes the case that "The mathematical accumulated knowledge is a scientific commons that should be preserved and made easily accessible for eternity". It discusses several initiatives, such as EUDML with that mission.
As far as I can tell, none of these repositories allows an author to submit errata on their own. For that purpose the route through arXiv seems the way to go for now. That repository is extensively mirrored and is committed to ensure its content will be retrievable a century from now. Since revised versions of a submission are linked to a common identifier, the second issue is resolved as well. Even if the paper is not yet on arXiv you can most likely post an author-prepared version without violating copyright. If it's a really old paper, this may mean retyping the text, but that may well be worth the effort.