The shady hat is a unique, fedora-style pre-War hat in Fallout 3.
The shady hat can be obtained as a reward for doing the mirelurk observation quest as part of Moira Brown's "The Wasteland Survival Guide" missions. Place Moira's observer unit inside a nest deep in the Anchorage Memorial. To get it the Lone Wanderer must place an observer unit in a mirelurk egg nest without killing any of the mirelurks in the area.
The shady hat can only be repaired with merchants.
Playstation 3 While wearing the Chinese stealth armor and the shady hat, changing both head and bodywear will yet retain the effects of the shady hat. [verified]