I believe what he was saying is the latter; his passport had expired or was lost, and so he needed a new one to travel to another country. With the Federal government shut down (including, presumably, those who issue passports) he was unable to acquire a new passport; the question of whether he'd waited until the last "minute" and gotten an expedited passport, or had ordered it with plenty of time is moot, as if it was not yet issued when the shutdown began, he would not be able to get it until after the gov started back up (which hasn't happened yet).
When he said it isn't ordinarily a problem, I took it to refer to the fact that you can normally travel all about the US (which is probably the vast majority of his travels) without a passport. But he could have also been referring to the fact that you can normally get a new passport issued relatively easily and quickly (well, it takes a while if you don't pay for expedited, but even then it is a matter of weeks).