Yes, it is possible to create a fully anonymous website. However, you'll need to be very vigilant about it. Each internet activity creates an information trail leading back to the person who initiated the activity. This trail can be traced by hackers, government agencies, and ISPs. As an anonymous website owner, it will be your job to anonymize all activities.
You can do this by minimizing the activities around your website and using a VPN to hide your IP address. Be careful about any social interactions as they may reveal personally identifiable information.