Suspiciously good accuracy using neural network
As you can see, there is nothing significantly separable, which really confuses me. I am definitely using the test set to run the cross validation which is 40% of the sample data.

Well, your interpretation is wrong: there is nothing  linearly separable on the two first directions.
The other components in PCA may be more relevant. Remember that PCA gives you the direction where the data has most variance, which doesn't necessarily mean it is the direction in which it is most easily separated/classified.
I see you tryed LDA, which follows on that assumption that the most variable direction is not the most significant for classification, but since it is linear we can conclude that your data is not Linearly separable.
Your model seem pretty large, it is considerably more powerful than LDA, LR, etc.
If you are not confident with the results, try using other validation methods such as K-FOLD or Leave-one-out.