Can we fully simulate molecular physics? Is our knowledge of physics complete enough to achieve fully natural simulations of molecular interactions in a computer simulation? How far off are we? 
Reason for question: I wonder how far we are from simulating cells in a computer. I assume that this would be possible once we have complete models for physical and chemical interactions. Once we can simulate cells in a computer and run hundreds of experiments simultaneously, our knowledge of cell biology could start to increase very rapidly, allowing us to for example discover medical cures faster. 
 A: Computational chemistry methods have advanced significantly in the last 5-10 years, including much more accurate DFT methods, quantum mechanical dynamical methods (like Car Parrinello MD, and better classical molecular dynamics techniques.
That said, dealing with the dynamics of molecular reactions is an active area of research. Perhaps the most promising technique at the moment is ReaxFF a molecular dynamics method capable of handling some reaction types.
There are several big unsolved problems.
The main one, I think, is that we have traditionally designed quantum mechanical and molecular dynamics methods to handle the most stable, ground state species. That is, by definition, not reactive species.
Another problem is sampling. By definition, reactions are rare events. So rare event sampling techniques are needed to properly handle the statistics without requiring huge amounts of simulation time.
But I don't think we need fully accurate methods to understand a lot about cells. As mentioned by other answers, there are more "coarse-grained" or continuum models that can tell us a lot too.
A: It depends on the level of theory you want to apply to a simulation. 
For example, the current state-of-the-art ab-initio calculations for a single low ernergy ($\lt 10~eV$) electron approaching and interacting with a molecule can cope with perhaps 20 to 40 electrons in the target molecule. Note that ab-initio calculations contain, in principle, no approximations. Thus, target molecules with 10 or more atoms are very difficult to calculate for this situation.
Many simulations of molecular dynamics can cope with 100s or even 1000s of small molecules, but they have two restrictions; they normally have simplified interaction potentials between the molecules and can run for relatively short timescales - perhaps in the picosecond range at most. So to simulate systems with thousands of atoms we need to make approximations. Molecular dynamics simulations can make many useful predictions of the behaviour of atoms and molecules on the molecular level - both physical interactions and chemical interactions, but they do this at the expense of making some approximations in the theory, so they are not ab-initio. 
To simulate a living cell at the moment would only possible if many approximations and simplifications were made. Many more would need to be made than in typical molecular dynamics simulations.
To give one example of the approximations that would be required to simulate a cell, consider protein folding. Proteins are synthesized in cells as long chains of amino acids. These chains spontaneously fold up into their normal functional shape. Understanding how proteins fold is a considerable challenge, and it is remarkable that proteins will fold into individual shapes which have very specific and useful functions to the cell. Thus to simulate an entire cell it would be necessary to make assumptions about how quickly proteins are produced and fold up. It would not be possible to calculate the folding of every individual protein in the cell because it is a hard enough challenge at the moment for us to calculate the folding of just one protein. 
So to calculate an entire cell is quite a challenge at the moment - and it is impossible from an 'ab-initio' physics point of view.
It is, however, an interesting challenge to consider - and some useful insight may be gained from the simulation of cell even though many approximations would need to be made to simulate it.
A: Why not remove unnecessary parts of the cell just for computational feasibility? Rather than simulate every atom in the cell, why not simulate everything within a small section of a single organelle on a single computer bank, having it interact with a series of other computer banks which simulate and react with equally tiny sections of organelles, while simulating only the organelle and the immediately surrounding matter. Using such a system, individual computers could handle a few atoms each, simulated to a very high level of detail, while each small section would constantly change, the handling of each atom being passed between the computers simulating the surrounding areas so that the entire organelle could be simulated seamlessly, if slowly. Quantum computing looks like a promising way to considerably increase the speed of this process should it become a more viable technology. I'm by no means the expert in this field, but I am quite good with computers and I've done a bit of studying on cellular biology and idk it should be possible but ludicrously expensive and impractically slow. 
