So first, there are two types of web storage, where data is stored locally in a user’s web browser. First, local storage, where the data stored has no expiration date and will persist after the web browser is closed. Second, session storage, where the data stored gets cleared after the web browser is closed.
So the data saved using web storage is stored locally on your computer. Each web browser has their own location to store this data. Looking at Wikipedia’s web storage page, it links to a few pages on StackOverflow (here and here) where others have listed the location of the saved data.