Are there open-access, free, non-topical academic data and research archives?
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.