For various reasons I'd like to store fresh meat and fish without freezing for as long as possible. I should preserve its original state as fresh and taste, so salting, cooking, vinegar is out of the question. I have this idea to put meat in glass jars (with empty space filled up with ghee or tallow to eliminate air which degrades meat too), and fish as whole fish (with skin and scales, I just remove the intestines and gallbladder before storage), in water-ice bath. The idea is to keep constant temperature just above freezing, i.e. around 0.5°C (33°F). I could set-up the ice bath in a double Styrofoam container and replace deep frozen plastic bottles with water made in the freezer daily. Would this keep my meat and fish in as close to fresh state as possible for longer then simply sitting in the refrigerator (where temperatures will vary by a few degrees due to the duty cycle and opening the door)? Is anything like this practised with success and how long would meat (specifically beef, lamb, goat) and fish (e.g. snapper, barracuda, salmon) keep fresh like this? 
Is it safe to store fresh meat and fish for a bit longer at 33F than at higher temperatures?  Probably.  OP's original question asked about two weeks, and the answer is: at 33F, food is probably safe for greater than two weeks, per the scientific literature on food safety.  I know everyone wants to quote FDA regulations or whatever here, but they are a one-size-fits-all oversimplification. I gave a summary of the science of different temperatures with storage life in an answer here.  Unfortunately, the website that was used to source a lot of that summary for that answer is no longer extant, as the author (a prominent food safety scientist named O. Peter Snyder, who actually came up with the principles for many HACCP protocols) died last year.  I suppose that information may be archived somewhere, but I don't have time to dig up another citation for it all right now.  One can find citations in food safety journals for various models and studies, but they often don't make it into consumer or business guidelines for food, as these are necessarily simplified. As I noted in the linked answer, pathogenic bacteria grow at progressively slower rates at lower temperatures.  Only a few types will grow at temperatures below 35F, and all growth should cease for normal pathogenic food agents by 29F.  At 35F, the safe storage lifespan is at least double that of 40F (a typical refrigerator recommended temperature).  At 33F, the literature would suggest that a 2-week lifespan for safety is perfectly reasonable. Of course, safety is not the only concern.  Food quality can degrade over time even if food doesn't become unsafe to consume.  The discussion in comments over aging of meat is a case in point: beef aging does introduce significant changes in flavor, texture, etc.  In cases of large aged beef cuts, many people regard the changes to be positive (after removal of surface mold, etc.), but the changes in fish over time may be less predictable without standard preservation methods. Removing air from the environment around the food will significantly help to maintain some freshness, but introducing ghee or tallow (as mentioned in the question) could change the products in unpredictable ways.  I should also clarify that it would be important to have sterile ghee or tallow.  Also, to use them for sealing will likely require heating.  Residual heat as the fat solidifies will cause thermal cycling, which could introduce additional problems (both from a food quality and a food safety perspective). Bottom line is that at 33F you're likely to grow spoilage microorganisms that will make the food taste awful faster than you'll grow pathogenic bacteria that will make you sick.  And yes, a consistent 33F temperature is likely to preserve food for longer than a less consistent refrigerator interior that likely cycles in a wide temperature range, hopefully below 40F (but frequently not; many people will be surprised if they start actually measuring the temperatures of things on their refrigerator doors).  As to how long you can store a particular food using your particular method before it degrades in quality - well, I'm not sure anyone can predict that. 