The classification of eye states (open or closed) has numerous potential applications such as fatigue detection, psychological state analysis, smart home devices controlling, etc. Due to its importance, there are a number of works already reported in the literature using traditional shallow neural networks or support vector machines, which have reported good accuracy (about 96%). However, there is still enough space to improve the accuracy of existing systems using proper classification methods. The major problem with traditional classifiers is that they depend on manual selection of features and that is very challenging to select meaningful features for such classifiers. Convolutional neural networks (CNNs) have become popular for computer vision and pattern recognition problems with better performance than traditional methods. In this study, we proposed a model of CNN (EyeNet) for eye states classification and tested it on three datasets (CEW, ZJU, and MRL Eye). The quality (or diversity) of a recently proposed larger dataset (MRL Eye) has been compared with two other existing datasets with respect to the sufficient training of the model. The model shows very high performance (about 99% accuracy for classification of eye states on the test set of data when it is trained by the training samples of the same dataset. The proposed model improves the accuracy of the best existing method by about 3%. The performance of the model for classification of samples coming from different datasets is reduced when it is trained with the MRL Eye dataset. This concludes that even though, the MRL Eye has a large number of samples compared to other datasets, but diversity still lacks in the MRL Eye samples to sufficiently train the model.