The Washington Redskins hold the longest running sellout streak in NFL history. The Redskins have sold out every single one of their home games since 1968.
There is a bit of a caveat with this particular information. The NFL is made up of what was once two separate professional football leagues, the AFL and NFL. The record, dating to 1968, is only including sellouts during the years after the merger. The Green Bay Packers have the longest sellout streak of any current NFL team, but their streak began before the two leagues were merged and became the NFL as its known today.