This is a proposal for a minimal filesystem API, designed for only the use case of sandboxed local storage.

Getting the root directory:

partial interface Navigator {
     Directory getSandboxedFilesystem();
};

The Directory interface:

interface Directory {
    readonly DOMString name;

    // The request will return the newly created FileHandle
    DOMRequest create(DOMString name, Blob blob);

    // The request will return the newly created Directory
    DOMRequest makeDirectory(DOMString name);

    // The request will return a File or Directory
    DOMRequest get(DOMString name);

    // Deletes a file or directory; DOMRequest returns boolean true or false to indicate success
    DOMRequest remove(DOMString name);

    // Atomically moves or renames a file or directory; DOMRequest returns boolean true or false to indicate success
    DOMRequest moveTo(DOMString or File or Directory entry, DOMString newName, optional Directory newParentDirectory);

    // The request will return a FileHandle; error if called on a Directory
    // If File is passed, it must be a descendant of this directory
    DOMRequest openForWriting(DOMString or File file);

    // The request will return a FileHandle; error if called on a Directory
    // If File is passed, it must be a descendant of this directory
    DOMRequest openForAppending(DOMString or File file);

    // The DOMMultiRequest returns File or Directory objects
    DOMMultiRequest readEntries();

    // The DOMMultiRequest returns File or Directory objects
    DOMMultiRequest readEntriesRecursively();

}

The FileHandle interface, representing a file open in read-write mode (for the read-only case, FileReader already exists, so why reinvent the wheel?)

interface FileHandle {
    readonly attribute File file;

    attribute long long position;

    // Result for all of these is the FileHandle
    DOMRequest readAsArrayBuffer(long size);
    DOMRequest readAsText(long size, optional DOMString encoding);
    DOMRequest write(DOMString or ArrayBuffer or ArrayBufferView or Blob value);
    DOMRequest truncate(optional long long size);

    DOMRequest flush(); // fsync

    void close(); // Currently outstanding operations run to completion, but no more may be issued
}

partial interface URL {
    /// Gets a persistent URL for a File, if it's part of a local filesystem; otherwise returns null
    DOMString? getPersistentURL(File file);
}

Result interfaces:

interface DOMRequest {
    readonly attribute DOMString readyState; // "pending" or "done"

    readonly attribute any result;
    readonly attribute DOMError error;

    attribute EventHandler onsuccess;
    attribute EventHandler onerror;
};

interface DOMMultiRequest : DOMRequest {
    void stop();

    // Call if all results have been delivered
    attribute EventHandler onfinish;
};