Best practices for backing up your data suggests maintaining three copies:
For more information and best practice information, visit UK Data Archive: Backing Up Data.
Inter-University Consortium for Political and Social Research (ICPSR) maintains a data archive of more than 500,000 files of research in the social sciences. An international consortium of more than 700 academic institutions and research organizations, ICPSR provides training in data access, curation, and methods of analysis for the social science research community.
Dryad hosts research data underlying scientific and medical publications. Historically, the repository has been strongest in the life sciences. Most data in the repository is associated with peer-reviewed journal articles.
Dataverse: The Dataverse Network is an open source application to publish, share, reference, extract and analyze research data. It facilitates making data available to others, and allows you to replicate others work. Researchers, data authors, publishers, data distributors, and affiliated institutions all receive appropriate credit.
Open Science Framework has many tools to store, organize, and share your data throughout the research data lifecycle. This free resource comes with features meant to streamline your workflow amongst project collaborators as well as make your work more discoverable. You can also register your work to freeze a version of your project.
re3data.org is a tool for helping people identify and locate online repositories of research data.
OpenDOAR an authoritative directory of academic open access repositories.
Open Access Directory's Data Repositories Wiki is a list of repositories and databases for open data.