What do you say to arguments about overfishing in the Great Lakes and oceans?  Isn't this a clear example of what Keynesians might call market failure, with huge, greedy companies "free" to catch all the fish while driving smaller fishermen out of business and depleting our natural resources?

It is woefully ill-informed for a Keynesian or anyone else to claim that problems arising from non-existent or ill-defined property rights arrangements in fisheries constitute market failure.  Markets rely upon private property, not socialized property, as is the case with "publicly owned" oceans, lakes or other waterways.  Problems that arise from such socialized property should be properly attributed to the absence of private property and the socialists and other assorted statists should have to grapple with them.
Nonetheless, I am a believer in the power of markets and their ability to make lemonade out of lemons or, in this case, to make some rational sense out of the socialists' "tragedy of the commons" that occurs with publicly-owned fisheries. 
Let me first suggest that it is ludicrous for anyone to argue that small firms are somehow unduly disadvantaged in the fishing business, even with the problems of socialized waterways.  Fishing is one of the easiest and most inexpensive occupations to enter.  I did it myself when I was a kid, with a K-mart fishing pole and nightcrawlers you don't even need a license or a zoning permit to dig for.  I sold bluegills by the dozens to friends and relatives and I didn't give a damn that Mrs. Paul's Fishsticks, Inc., had a bigger bottom line than me.
For more on why free markets do not tend toward monopoly, check the archives for our "Ask the Economist" feature on our web page.
If markets are "overproducing" (that is, "overfishing"), it's purely because of the lack of property rights and the incentive structure this creates.  If no one owns the waterways (or "everyone" owns them), then there is every incentive to use the resources and leave nothing behind.  If people cannot claim either fish or where the fish live as their own private property, then they have no incentive to preserve the capital value of the resource itself.
This problem is not present with, for example, cows.  If cows free ranged across public property and no one could fence any of it off and "raise" them, then we'd all be shooting cows and nobody would be producing new ones.  We'd have a steak shortage in no time, which is precisely what happened to the buffalo in the American West in the last century until private property rights were permitted, encouraged, and enforced.
This is not a problem now with fish either, in the case of private fisheries that are producing an increasing share of the fish, shrimp, and other "seafood" that Americans eat.  Fish farms are all over the place.  Most catfish you eat in restaurants in the South come from private fish farms. 
Now, it isn't so readily apparent that the problem we've been discussing here can be solved even in large bodies of water like the ocean.  However, "aquaculture" experiments do exist and have met with success in parts of the ocean off our coasts.  Technology is likely to make it all the more feasible over time, and infinitely more creative and productive than decrees from bureaucrats.  This is not an area where I am an expert, but I would like to call your attention to the Political Economy Research Center in Bozeman, Montana.  PERC produces studies and commentaries that address issues just like this one, and I am sure they can provide you with real-life examples.
Incidentally, most of our home lighting once came from whale oil--up until the mid-1800s when the socialized whales living in socialized oceans grew scarce.  Thank God we didn't try to "solve" the problem with a new Department of Whales.  The price of whale oil rose and markets responded by finding an alternative--namely, crude oil.  I mention this not because it directly answers your question, but only because it dramatizes the fact that markets are wonderfully good at solving the problems of socialism in ways that lazy and uninspired socialist intellectuals would never have an ounce of brain power to come up with themselves.
When you run into a Keynesian or other statist who poses the problems you mentioned in your questions, I suggest you advise him that private solutions can be found in probably half the time he spends concocting silly arguments for why the problems of socialized property are the fault of the market. 

"Markets rely upon private property, not socialized property as is the case with 'publicly owned' oceans, lakes, or other waterways."

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