All containers are a pain to use. But I tend to use pots for meat as well, because they are very convenient.
You need a food stockpile that accepts pots. It is almost always best to make this stock pile *only* accept meat (or meat and fish, if you also want to put fish in pots). Set the number of barrels to the max, but keep in mind that no matter what settings you set, it will always reserve 1 space for an empty barrel (I think this is a bug).
Now the secret to understanding barrels (and other containers) is that dwarfs will carry to the barrel to the item, fill the barrel *with a single item* and then return the barrel. So if you have your trade depot full of meat and it is 40 tiles away from the meat stockpile, your dwarfs are hauling barrels back and forth *for each bloody piece of meat*. Especially since a rock pot can contain something like 60 pieces of meat, your poor dwarf is going to haul that damn thing back and forth to the trade depot 60 times! i.e., 4800 tiles hauling a heavy barrel -- way more than enough time for most of the meat to go off.
So the trick is to make an intermediate stockpile that for meat (and potentially fish, or anything else you want in a barrel). Put this stock pile near your permanent stockpile, but *don't* allow barrels. That way multiple dwarfs can all grab a single piece of meat or fish, and deposit it in the intermediate stockpile. In the meantime another dwarf will grab a barrel, haul it to the intermediate stockpile, add a piece of meat and then haul it back again.
The intermediate stockpile needn't be large. Just big enough to deal with the difference in speed of delivery. It's also handy because it is a visual indicator when your barrels are full, because you'll have some spilling over into the intermediate stockpile. If *that* one is full, then you might consider dumping some meat so that it spoils outdoors.
Hope that helps!