This is not an exhaustive list; My hope is it gives you some ideas of where you can find data sets.
FedStats The gateway to statistics from over 100 U.S. Federal agencies.
General Social Survey Data and Information Retrieval System (GSS-DIRS) The GSS is widely regarded as the single best source of data on societal trends.
Inter-University Consortium for Political and Social Research The world's largest archive of digital social science data.
Roper Center for Public Opinion Research One of the world's leading archives of social science data, specializing in data from surveys of public opinion.
Cornell Institute for Social and Economic Research, Internet Data Sources for Social Scientists An informal list of sites that CISER staff have found useful.
Bureau of Justice Statistics Includes statistics about: crime & victims, prosecution, federal justice system, criminal offenders, law enforcement, and more.
National Archive of Criminal Justice Data Provides criminological and criminal justice data collections on specific topics.
National Institute on Aging Conducts research on aging processes, age-related diseases, and special problems and needs of the aged.
AARP Research Center Features authoritative information on issues affecting the 50+ population.