Measuring American Legislatures
The Measuring American Legislatures project is an effort to code the ideological orientations of both state legislatures and individual state legislators. Data from the project are available via the Dataverse of Boris Shor, who is one of the project's investigators.
National Institute for Money in State Politics (NIMSP)
NIMSP is a watchdog group with an extensive clearinghouse of data on campaign contributions and lobbying, with a focus on elections at the state level. Bulk downloads of data are also available via http://sunlightlabs.github.io/datacommons/bulk_data.html.
Partisan Balance of State Government
Carl Klarner at Indiana State University has assembled various datasets on state politics, including data on outcomes of state elections and partisan control of state legislatures and executives. You can also access Klarner's data via his Dataverse.
State Ideology Data
The State Ideology Data, which have been assembled by Richard C. Fording at the University of Alabama-Tuscaloosa, measure the ideological leanings of both the public and political leaders for the individual states. The data cover the years 1960 onwards.
State Politics and Policy Quarterly Data Sources
This site has various collections of state-level data for the United States. The Practical Researcher Data provide time-series data on political, judicial, economic, and social variables. Depending on the variable, the data coverage is from 1975 to 2006. There is also a collection of individual datasets on particular topics such as partisan balance of state governments. SPPQ also has a journal replication archive with other data relevant to the study of state politics.