The Smithsonian Institution offers a multitude of internships in various museums and offices. Some of the internships have a small stipend. There also are opportunities to hold a “virtual internship” that would not require relocation to Washington, D.C.
There are many internship and employment opportunities for undergraduate students at the Library of Congress. Of particular note is the Junior Fellows Program.
The U.S. Department of State offers paid and unpaid internships for college students. The positions might be in Washington, D.C. or at U.S. embassies around the world.
The Congressional Hispanic Caucus Institute, Inc. offers paid internships for Latino students in Washington, D.C.
Historical societies and living history sites that offer internship programs:
Newport Historical Society – Buchanan/Burnham Internship
“Virtual” Internships that do not require relocation:
Archives and Presidential Libraries
Most of the presidential libraries and archives offer some form of an internship or student volunteer program. These vary in requirements, work performed, and pay status. See Presidential Libraries Internship Opportunities or the specific websites for details.