You can get 7 days of data from the Search API, or you can listen on the Streaming API for an indefinite period, for up to 1% of the firehose.
For more complete datasets you can use Gnip, which is a commercial offering based on the full historical Twitter archive.
We have previously offered a limited Data Grants program for academia, and are looking at future possibilities in this space, but right now these are the only two options available.