Do barrels and other containers degrade with use in Dwarf Fortress?

There is no degradation mechanic for barrels. What tends to happen is that dwarves aren't the most organized race, and so they have a bad habit of putting just a single seed in a barrel, or leaving one cracker at the bottom. The brewers are lazy, so they'll only use a completely empty barrel. One thing that can help is to limit the number of barrels in your food stockpile, which'll prompt the dwarves to fill up the existing barrels rather than grabbing a new one, and any overflow will be set on the ground in the stockpile.

In addition to what SaintWacko said, you can get some weird job cancellations if two dwarves happen to try to interact with the same barrel at similar times. Dwarf A decides to go get a barrel. Meanwhile, Dwarf B also wants a barrel, coincidentally picks the same barrel as Dwarf A, and gets there first. Dwarf A will then cancel his job message.

Barrels, well anything really, can also decay if left in a stockpile that also accepts refuse.