Many funders and journal publishers require researchers to publish their (anonymised) research data in an open repository unless there are ethical, legal or commercial reasons for not doing so. The general principle is to make your research data as open as possible, as closed as necessary. Not all repositories have controls to support restricting access to datasets. If you have qualitative research data, please read this guidance on Making qualitative data re-usable. Three types of repository are described below. Please contact the Research Data Manager, Nicholas Syrotiuk, if you have a query about data repositories or publishing research data.
Durham University has it's own research data repository.
It's available for all staff and postgrads at no charge. We currently accept deposits up to 700 Gb. You will receive personalised support. See: Guidance on publishing research data in the University's data repository.
All of these are well established and widely used.
Provided you find a suitable repository, this is a good choice because others might find your research data by browsing. Try finding a subject-specific repository.
Some examples: