In the year 2448 from Creation (1313 BCE), Tammuz 16 was the 40th day following the Giving of the Torah at Mount Sinai, and the people of Israel wrongly expected Moses' return from the mountain (he would actually return on the following day). When their leader failed to return, they demanded from Aaron: "Make us a god that shall go before us". Hur (Moses' nephew, the son of Miriam and Caleb) tried to stop them and was killed by the mob. Aaron fashioned a calf of molten gold.
Links:
The Making of the Golden Calf (text of Exodus 32 with Rashi's commentary)
An anthology of Midrashim and Commentaries on the making of the Calf
The Day Before (on the deeper significance of Tammuz 16, from the teachings of the Lubavitcher Rebbe)
More on the Golden calf
See also "Today in Jewish History" for tomorrow, Tammuz 17.
It is not so much that we need to be taken out of exile.
It is that we must take the exile out of ourselves.
What is exile? It is the belief that there is some place outside of G-d.
And that in that place, there are forces stronger than G-d.
And those forces will not allow you to do what your soul came here to do.
Fear no thing, no force, no being other than G-d. Do that which you must do as though nothing stands in your way.
Then you will have taken yourself out of exile. And G-d will complete the job.