The flat file database that wouldn't die
If you think getting things to change in the military's infrastructure is bad, try the true black hole of U.S. government IT: the IRS. For decades, all of the IRS's tax records were held in something called Individual Master File, an enormous flat file containing millions upon millions of records, stored on big spinning wheels of magnetic tape and accessed via COBOL code. This file was the cutting edge of tech sophistication in the 1960s when it was first implemented, but fifty years later it had grown a little rough around the edges, yes?
Ha, we kid about the 50 years thing: it was outdated by the '80s, by which time most everybody else had migrated to relational databases. Still, Master File managed to defeat all the billions of dollars and replacement projects the government could throw at it, trudging happily along and restricting the IRS from accessing individual files more often than once a week. It was only this year that Master File was replaced by Customer Account Data Engine (CADE), a more conventional database system run on IBM hardware -- CADE 2, actually, since the original CADE was a failed project that was killed after years of development. And even that switchover was only for data for individuals; taxes for businesses and retirement plans are still kept on the old system for now.
Big Blue predicts blue skies
Sometimes longtime government use of technology isn't the result of chaos and incompetence, but simple thriftiness. When it comes to weather predictions, satellites and doppler radar get all the press, but the National Weather Service still relies on good old-fashioned balloons -- and much of the data sent from those balloons is still processed by good old-fashioned IBM PC/XT machines, dating from the 1980s. It did turn out that 640 KB wasn't enough for everybody, but it seems that it's still good for many purposes.