"start storing" is correct and usable in written English.
You can use it in a number of contexts to mean beginning the process of organizing, or safeguarding something. For example, "It's time to start storing the old documents in the archives."
In times of crisis, he said, "people think it is the end of the world and they start storing stuff that should not be stored".
Financial security will improve as the 2 billion people who do not have a bank account start storing money and making payments using mobile phones.
By the end of the day, he had ordered crews to start storing some trains underground for the night, a standard procedure to keep subway cars from freezing up.
The United States' government's advice isn't especially encouraging: In a remarkable speech over the weekend, Secretary of Health and Human Services Michael Leavitt recommended that Americans start storing canned tuna and powdered milk under their beds as the prospect of a deadly bird flu outbreak approaches the United States.
So why do we start storing data?
When prices go up, people figure they better start storing foods and precious metals.
"Habitat started storing stuff in 1985 and got 15 years of free space," Mr. Hayduk said.
Lately my wife has started storing dahlia tubers and other flower bulbs in her box for the winter.
I had postnatal depression and started storing up my antidepressants so I could do away with myself.
I've already lost pictures, videos and audio because of lost and broken drives and computers, so I've started storing media in the cloud more and more.
Although Twitter inked a deal with the photo-storage site Photobucket in June, the company has since started storing images on its own servers.
