Are there open-access, free, non-topical academic data and research archives? I am looking for an archive where anyone can deposit both data and research papers, which does not require creators or users to pay, does not require registration for users to access materials (hence not academia.edu), and which is not restricted to specific fields such as math or computer science (though the content could be limited to academic stuff as opposed to recipes). If that's not enough, the web site should provide "adequate amounts of space", which I know is field dependent, but let's start with 250 GB per user as a baseline. I presume that creators of content to be posted on such a site would have to register. As far as I can tell there is no such archive, but I may have missed something.
In addition to the already mentioned figshare service, I would recommend you to take a close look at Zenodo service. While Zenodo also offers a set of features, similar to figshare, including an automatic DOI generation and assignment, there are at least three significant differences (and Zenodo's advantages, that is, for the appropriate cases) that ought to be considered, as follows:


Zenodo offers git and GitHub integration, so, if you plan to publish your research source code, I would go with Zenodo (and I actually did with my dissertation research source code - it's a bit tricky, but nothing that cannot be solved);
Zenodo is backed by world-famous CERN and its solid IT cloud infrastructure, so your data is IMHO more safe than with figshare in the long term.
Additionally, from an organizational perspective, a related argument is that Zenodo backed by a stable organization and European Union's multi-year projects (and, thus, funding), whereas figshare, while an independent organization, is backed by two not very well-known private (and, thus, not so stable) companies.