The Museum of the Institute of History and Philology, at Academia Sinica, has an excellent collection of rare texts and artifacts including Han Dynasty wooden slips from Edsen-gol and numerous rare texts. The collection came to Taiwan when the Academia Sinica Museum was relocated in 1949, See more about the history here.
National Central Library has its origins in Nanjing, where it opened to the public in 1936. As the library notes on its webpage, its special collections, “range from bronze and stone rubbings, Dunhuang scrolls, rare and exquisite Chinese books made before 1795, and thread-bound volumes produced between 1796 and 1911. The rare books amount to approximately 12,900 titles and more than 130,000 volumes from the Song (960–1279) to Qing (1636–1912) dynasties, especially with a rich anthology of Ming prints. The thread-bound books total more than 9,700 titles and 110,000 volumes.” See also National Central Library’s Taiwan Memory page for books, postcards, photos, local history, contracts, etc.
The National Palace Museum is famous worldwide. For more on the history of the collection, read here. For more on the National Palace Museum databases and indexes, see their entry in the ACROSS site,
Ri Xing Typography (日星鑄字行) has the “last complete set of traditional Chinese character molds for lead-type casting in the world.” In addition to the Taipei Times article linked above, which contains Ri Xing Typography’s address, more information on the factory is available in this Focus Taiwan article and a post in the blog City543.
Suho Memorial Paper Museum provides information on the history of paper-making in Taiwan, including the making of pith paper.
The Museum of Church History holds archival material related to Christian missionaries in Taiwan. An appointment is necessary to visit.
The National Museum of Taiwan Literature was founded in 2003. In 2013, several items in the collection were recognized as “Significant Antiquities” according to the Cultural Heritage Preservation Act.
I hope to continue to update this page. If you know of a site that deserves mention, please let me know in the contact form.