Funders may specify a repository for deposit of publications. For example, the NIH policy specifies deposit of publications to PubMed Central.
N.B. Most funders require deposit of a publication upon acceptance, not upon publication. Embargoes (delayed access) can be supported in most repositories if expected by your publisher. Also note which version of the article your funder requires you to deposit. A post-print is the same as the final manuscript. It is the final version of your article after peer-review but before the publisher has formatted it with page numbers, watermarks, and the like. The publisher's formatted version is known as the publisher's version.
For those policies that don't specify a repository, the Directory of Open Access Repositories, known as OpenDOAR, provides a long list of repositories for disciplinary research. Examples of disciplinary open access repositories include:
Institutional (as opposed to disciplinary) open access repositories are also available. At UT, our institutional repository is Trace -- Tennessee Research and Creative Exchange. Preserving and archiving research for years to come is part of a library's mission, so please deposit a copy of your articles in Trace, even if they aren't subject to a funder's public access policy.
For help with retaining the right to deposit your work in Trace, please see the page on Publication Agreements from the UT Libraries' Scholarly Publishing Toolkit.