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:
GitHubintegration, 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
figsharein 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.
Figshare sounds like what you're looking for.
You can upload papers, data, figures, code, and/or other stuff. Another nice feature is that you can get a DOI for your uploaded stuff.
A free account (for content creators; consumers don't need an account) includes unlimited public storage.
I believe Figshare may now do what you need - it will take papers as well as datasets (and other outputs).
For now there is a file size limit of 250MB with our standard figshare accounts alongside 1GB private storage and unlimited public storage.
...so let's call that 'maybe', depending on your exact definitions.
A caveat - if you mean depositing already published material when you say "deposit research papers", you may have problems whatever service you use. This is primarily governed by your existing agreements with the original journals, which in most cases are quite restrictive. Caveat emptor.