Secondly, how would such an archive work? Who would pay for it? How would it be safeguarded in such a way that it can be read by 'far future anthropologists' but not the people paying for the storage?
If we're only talking about public chat rooms, it shouldn't be difficult to archive the content of those.
There are open repositories of the entire internet text content (common crawl). These scrapes are periodically repeated. That's orders of magnitude more data than all discord messages ever.
So technically it's not a problem making such an archive. The financing is of course always an issue, but not because the costs are large.
Secondly, how would such an archive work? Who would pay for it? How would it be safeguarded in such a way that it can be read by 'far future anthropologists' but not the people paying for the storage?