Libraries of big universities could simply provide the infrastructure to publish (online only) journals. There is not much needed as most of the work is already done by volunteers (reviewers / editors) so this could be really cheap.
The NIH, through PubMed Central, already provides the infrastructure for archiving (biomedical) articles. In fact, they demand that any publications resulting from NIH-funded research be archived there (with a 1-year delay from release by the official publisher). I believe ERC has a similar requirement for European research Some of the best journals have put their entire historical archives there (J Physiol back to 1878), but most journals only since the 2008 NIH mandate.
The problem is not the infrastructure to do online publishing. The problem is incumbency. The people who actually run the journals are, for the most part, tied to their historical publishing partners. I'm thinking especially of the 'big' journals that are the official publications of various academic societies. They are as locked in to publishers' workflow software as most people are to Microsoft Office.
Personally, I think every academic society, each of which claim education and public dissemination of science as core values, should make their historical archives available through PubMed Central, arXiv, or similar. Most of them have been digitized. Most of them are available to society members or journal subscribers. Most of them cost $30-$50 per article for the public to read, and there's no reason for that. If the society you belong to has not released its legacy content, ask your leadership, Why not?