Is Schrodinger's Cat a real conceptual problem or just a problem with approximations? In this thought-experiment a cat is placed in a box set with a bottle poison that will release and kill it depending on whether or not a certain radioactive particle decays. The box is kept closed and it is asked "is the cat alive or dead?" Since the decay of the particle is a quantum mechanical process- it is represented by a state function. This function remains in a "superposition state" until it is observed whether the particle has decayed or not. The cat is presumably and unfortunately entangled with this function- it's own life/death is in a superposition state until it is indeed observed to have dodged the bullet or not. Schrodinger used this thought experiment in an attempt to show the ridiculousness of certain aspects of quantum theory. His motives were clear in devising it and are well documented in his correspondences with Einstein. 
My question is not about the cat, the particle, or Schrodinger. I want to know about the box. The box that contains this scenario must effectively isolate the outside observer from any information about what is (or could be is perhaps more correct) going inside of it. So what sort of box would qualify? It seems that such a box could not have any interaction at all with the outside universe- no gravitational or thermal effects- anything. Otherwise would not some kind of information about what was going on inside escape without it being opened?
And this being said- were such a box constructed would it not be entirely cut off from reality and how could we be sure it even contained a cat still or was even there at all?
Is this experiment possible or are the problems I outline above a deterrent to it's physical possibility?   
 A: You're right that this experiment isn't really a realistic one. In fact, it's difficult to imagine how we could even tell whether a given box were such a perfectly isolating box.
However, if we put aside the (somewhat important) question of empirically determining whether we actually obtained a cat in a superposition of life and death, we can try to imagine the circumstances under which the superposition of life and death would decohere very slowly. That is, by considering a nearly isolated system which would almost evolve exclusivey to the Schrödinger equation.


*

*Fortunately, gravity is a very weak influence. Assuming that superpositions are not themselves inherently sensitive to substantial differences in mass distributions for reasons other than gravitational potential, it suffices to restrict the cat's motions in such a way that its ability to move while alive does not substantially affect the gravitational forces on its environment. That is: allow the cat much less room to move than the spatial separation of it from objects outside of the box. For instance, by putting it in orbit, or perhaps deep space.

*The cat's motions shouldn't leave any hints in the motion of the box about the box+cat centre of mass. So you should probably make the box very massive, and perhaps a smooth metallic spherical shell for good measure, so that the surfaces of the box give very little away.

*Finally, the inside of the box should be reflective on any wavelengths with which the cat is likely to interact in the electromagnetic spectrum, i.e. in the high microwave through ultraviolet range of the spectrum.
Just how quickly the wavefunction of the cat would decohere I could not say; it would be sensible to try and discern this by considering, for a test system, how quickly its position or energy levels would become discernably different for different possible variations in the mass distribution and light-reflections/emissions of the box. This would allow you to determine the half-life of the coherence of the cat's wave-function — in principle, of course, because as soon as you try to draw closer and observe the cat, that very interaction will cause its wavefunction to decohere very quickly, and more quickly the more acutely you try to discern its behaviour.
The real test of the Schrödinger's cat experiment, barring any fundamental advances in theoretical physics, would be to invent a machine which allows you to re-interfere the two components of the wave-function to get a cat which is definitely still alive. However, this (a) is something of a pipe dream, as it would amount to overcoming the second law of thermodynamics; and (b) would be experimentally indistinguishable from succeeding in building a box in which the radioactive-poison-trigger doesn't work.
A: Your question effectively reduces to that of where you place the Heisenberg cut in an idealized model of the experiment. That's not different from the issues that are already probed by the original thought experiment and the Wigner's friend variant.
It's implicit in the quantum formalism that (unless one introduces POVMs) one models experiments using constructions of the form $\mathsf{Tr}[\hat\rho\hat O]$, so a quantum model must distinguish what aspects of a set of experiments are to be modeled by quantum states and what aspects are to be modeled by measurement operators. The decision, effectively the decision of where to place the Heisenberg cut, is made on the basis of where there is a relatively small degree of entanglement between the measured systems and the measuring systems. In principle there is no perfect place to put the Heisenberg cut, but for practical purposes there are better and worse choices. Saying that the box isolates the cat from the outside world is just to say that, given the experimental apparatus available to the experimenter, whether the cat is alive or not is indeterminate. We can make that not true either by making the box less isolating, so that less sophisticated measurement apparatus can discern whether the cat is alive or dead, or we can introduce more sophisticated measurement apparatus, so that we can tell despite the precautions taken by the makers of the box whether the cat is alive or dead.
The introduction of POVMs, BTW, can be modeled just by introducing additional degrees of freedom, called ancilla. The box is effectively an ancilla system for the cat + radioactive isotope system, whatever measurement apparatus we might introduce may be modeled as a further ancilla, etc., but we can also use POVMs to take account of whatever entanglements there may be between various parts of the experiment. This provides one way to talk about different levels of models of experiments in a mathematically coherent way.
I like the Answer from Niel de Beaudrap, but I'd almost finished this when his appeared.
A: Depending upon your position on the black hole informational content of Hawking radiation, a black hole could be such a perfect box. Unfortunately, some people like Susskind claim it is too good a black box that by black hole complementarity, the interior of a black hole and any cat within it doesn't exist. But what does it mean for a cat to exist? Here comes the philosophers waiting for a chance to bite.
