Can you rephrase this sentence (about storing files)?

It is certainly possible.

"Starting from this date the storing of all such files will be in folder B."
"Starting from this date the storing of all such files will be relegated to folder B." "Starting from this date the storing of all such files will be specified for folder B."

However, all of these are awkward and less understandable than the original phrase. It changes the voice from active to passive, by making the verb "stored" into a gerund noun "storing". (Not it's not actually passive voice, but it's certainly a more passive phrasing.)

I don't think it is possible without creating an ugly or ungrammatical sentence. The subject of the utterance is the storing, which must be modified by an adjectival phrase meaning will be performed using folder B for storage. Not easy to do without repeating oneself.

The original sentence is brief and precise. The only improvement I can see is that I personally would change Starting to As, and will be to are. Where there's no loss of clarity or accuracy, short words are always preferable - particularly in technical manuals & user guides.

I think this problem arises because logically the real subject of the utterance should be all such files, for which the attribute being specified is where they are stored. Introducing the storage (I don't like storing at all here) as an intermediate noun phrase simply makes the whole thing semantically and grammatically convoluted.

IMO, a native English speaker would say "Beginning on [give the actual date, not "this date"], the system will store all such files in Folder B."

-- pete

Starting from this date all such files will be stored in folder B.

There is nothing wrong with this.

Beginning on this date all such files will be stored in folder B.

Nothing wrong with this one either.

Nevertheless, I would probably phrase the sentence this way:

As of this date all such files will be stored in folder B.