Downloading file directly to user's file-system
If you by save/store mean directly to user's computer/file system then no, this is not possible due to security reasons.
However, you can initiate a download which will pop up a "save as" type of requester and allow user to accept or deny to download the file.
There are more than one way to initiate a download. An interesting one is the new download attribute for the anchor tag where you can specify a file name as well as automatically initiate the download setting its href to a data-uri or something else you want to reference for download:
<a href="urlToFile" download="myFile.ext">Click to download</a>
Local storage mechanisms
If you simply want to save the file locally you can use one of the many local storage mechanisms such as:
Note that all of these as sand-boxed and only available in the browser using the same origin as they was written from. The data may or may not be discarded at any point as well (by user or by browser) so they are not a "safe" storage (always keep a server copy or a way to regenerate the data).