localStorage is not deprecated, it's an integral part of the web platform. The quota for web storage depends on the amount of free space on the disk, so maybe you have none. It may be also a bug in Chrome, you can report it in
https://crbug.com.
> store small media resources (.png, .mp4) is recommended
The best is IndexedDB because it can store Blob or ArrayBuffer for the file contents directly. The API for this storage is complicated, but if you only use the basic get/put/delete methods then it's pretty simple. There are also many js libraries that offer a simplified Promise-based API for IndexedDB.