
It's probably true that you only get that cardboardy taste if you grossly oxidise the beer by letting air glug in. The yeast will devour a certain amount of oxygen, as will other reactions. But the taste will still change even if the oxidation is at too low a level to taste cardboardy. A general dullness of taste and progressive sourness is characteristic of this, although stronger ales tend to stand up better.
The rate of oxidation is the key issue and larger barrels, even if made of wood, could be fine simply because of the large volume of beer relative to the surface area. With a 5-gallon batch it's a little more challenging, although keeping the vessel full helps. Possibly, if the oxidation rate is low enough the beer may actually benefit (like maturing a wine) but I've never experienced this, maybe because my barrels never stay completely full for very long.
Or maybe you've got really good barrels that keep oxygen at bay for at least 6 months. If so, I'd like to hear more...