Is data visualization a sufficient indication for the separability of the data? What are other indications of data separation? In other words, let's say we have a data representation as in the image below, which is generated from the PCA, the projection of the data onto the first two PCs. As it's shown in the 2-D space, the data is not separable. Is that a good indication that this data is not separable? What other metrics can be used to figure out data separation?

 A: There is an asymmetry worth noting here. 
If a PCA plot shows distinct, separated clusters, then it is clear evidence for the separability of the data. But an absence of this kind of structure in the PCA plot (such as in your example) is not evidence for a lack of separability. 
This is because (as pointed out in the comments) your 2-dimensional plot above omits information in the dataset, assuming it contains >2 dimensions. You may simply be looking at the wrong dimensions! There is no rule that says that the data pattern or structure you are interested in must show up in the first two principal components; they are merely the dimensions with the most variation in the dataset. It is entirely possible for dimensions with less variation (i.e. principal components 3, or 4, or whatever) to be the ones on which the data are clearly separable. 
If you are interested in separability and identifying the dimensions that contribute to this, PCA might not be the most useful tool. As suggested by @naught101, a clustering approach is likely to be more useful. 
