This question asks about https://cogsci.stackexchange.com/questions/3933/psychological-journal-that-focuses-on-publishing-interesting-psychological-datas journals where you can publish psychological data. Such journals rely on other organisations to provide repositories for the published datasets.
In particular, the journal Open Psychological Data has a page which lists several data repositories along with an excellent summary of important features of each repository (e.g., location, focus, cost, licences, identifiers, sustainability, and depositing instructions). In case the link goes down, hear is summary of listed repositories:
