Do you need a dataset, on your topic or maybe just to practice on? See this NC State page (link below). Google has a newish search for datasets. I've had good luck with Figshare and ZENODO. Also, the Library is a member of ICPSR, which is good for the social sciences.
2025 -- Some government hosted data sets have been removed from their .gov locations. See Data Rescue Project's list of lists of archiving sites. Ask us for help if you still cannot access a data set.
Where are other researchers getting their data? Articles' methodology sections may lead you to publicly available data. In some cases, the researchers have measured phenomena and present original data.