You can share your data easily by emailing it to requesters, or posting it to a website, Google, Amazon or Microsoft. Unfortunately, this method of sharing makes it difficult for people to find your data. Depositing your data in an archive will facilitate its discovery, preservation, and citation.
The Online Registry of Research Data Repositories is a good place to start. Search it by topic or Browse by subject, content type, or country.
Caltech CODA - The repository created to capture, distribute, and preserve digital products of Caltech faculty and researchers. Authors can archive their digital works in a variety of formats, including datasets. Caltech CODA has been in operation since 2000 and has an institutional commitment through the Caltech Library to provide storage, preservation, and distribution services on behalf of the Caltech community in perpetuity. For more information on how to deposit data into Caltech CODA, contact coda AT library.caltech.edu.
Barcode of Life Data Systems
fMRI Data Center
Gene Expression Omnibus (GEO)
Protein Data Bank - An information portal to biological macromolecular structures
SumsDB - SumsDB is a repository of brain-mapping data (surfaces & volumes; structural & functional data) from many laboratories.
UniProt - Submit a new protein sequence to UniProtKB using SPIN, a web-based tool for submitting directly sequenced protein sequences to the Universal Protein Resource (for new nucleotide submissions, use EMBL's WEBIN instead).
XNAT Central - Central is a database for sharing neuroimaging and related data with select collaborators or the general community.