Sites like Mega.co.nz do such things.
The message you are seeing pertains to FileSystem API. As websites are becoming web apps, it is becoming necessary for them to act like actual apps. One such behavior is storing large data and settings files on your computer and access that data freely, reading, writing and changing them as opposed to a simple one-off download for you. That's why FileSystem API is invented.
In case of Mega.co.nz, it downloads an encrypted copy of file to the local storage, decrypts it and then lets you have the decrypted file in your Downloads folder.
Now, the purpose of this permission request: Would you want a web app that you don't even know – let alone trust – take up potentially hundreds of megabytes of your hard disk free space? That's what this permission request is for.
Google Chrome, which is the only browser that implements this API, has a cap for the allotted space, but one person might visit hundreds of domains each day. Thus, the impact would still be significant.