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Kaggle contains data extracted from social media platforms.
Dataverse includes social media datasets contributed by academic researchers.
ICPSR has collections of social media data in its archive.
Apply for Data Access for Academic Research
Accessing Twitter data for analysis depends on your question and time period. For Twitter searches (keyword or hashtag), data older than around seven days typically cannot be retrieved. You can collect data going forward in time. If you need to extract tweets from an account, you can collect data going back and going forward in time. Twitter's Developer Agreement and Policy has details about what you can do with the data.
A project by University of Illinois Technology Services and the National Center for Supercomputing Applications (NCSA) with the goal of making social media data, analytics, and visualization tools accessible to researchers and students of all levels of expertise. You will need to set up an account.
Developed by George Washington University Libraries, SFM was designed to help researchers extract data through Twitter, Flickr, Sina Weibo, and Tumblr. SFM is available through the Mason Libraries' Digital Scholarship Center.
NCapture captures data from Twitter, YouTube, LinkedIn as well as Web pages and online PDFs. You need to use NCapture with NVivo in order to extract the data. The DiSC Lab in Fenwick Library has NVivo installed on its computers.