Vision Transformers have been rapidly uprising in computer vision thanks to their outstanding scaling trends, and gradually replacing convolutional neural networks (CNNs). Recent works on self-supervised learning (SSL) introduce siamese pre-training tasks, on which Transformer backbones continue to demonstrate ever stronger results than CNNs. People come to believe that Transformers or self-attention modules are inherently more suitable than CNNs in the context of SSL. However, it is noteworthy that most if not all prior arts of SSL with CNNs chose the standard ResNets as their backbones, whose architecture effectiveness is known to already lag behind advanced Vision Transformers. Therefore, it remains unclear whether the self-attention operation is crucial for the recent advances in SSL - or CNNs can deliver the same excellence with more advanced designs, too? Can we close the SSL performance gap between Transformers and CNNs? To answer these intriguing questions, we apply self-supervised pre-training to the recently proposed, stronger lager-kernel CNN architecture and conduct an apple-to-apple comparison with Transformers, in their SSL performance. Our results show that we are able to build pure CNN SSL architectures that perform on par with or better than the best SSL-trained Transformers, by just scaling up convolutional kernel sizes besides other small tweaks. Impressively, when transferring to the downstream tasks \texttt{MS COCO} detection and segmentation, our SSL pre-trained CNN model (trained in 100 epochs) achieves the same good performance as the 300-epoch pre-trained Transformer counterpart. We hope this work can help to better understand what is essential (or not) for self-supervised learning backbones.