We call such institutions "safe deposit boxes." There is no version of a true equivalent to a safe deposit box that is widely available to the public and accessible via an electronic transfer in the United States. And, among muggles without access to Gringotts Wizarding Bank, very few people manage their cash savings that way. In part, this is because this children's book financial institution runs in a way that children and other people who don't understand banking think that banks work, even though they really don't work that way (and ultimately can't work that way at a macroeconomic level).
In many countries outside the United States, government run postal service banking systems with savings accounts from which you can make payments via a money order like document sometimes called a giro serve this purpose. Giro based payments through a postal banking system was the main means by which consumer payments were processed, in the era when U.S. consumers made payments with checks drawn on commercial bank deposit accounts, in much of the world. Some of these systems now allow for electronic fund transfers. The first postal banking system was established in Great Britain in 1861.
In Japan, for example, where this system was established in 1875, there have been times in recent history (I don't know if it is still true) when most household savings were in the form of deposit accounts with the postal system. As of March 3, 2006, the Japanese people had "about ¥330 trillion in assets, which is roughly 65% of Japan’s GDP" in aggregate postal savings account balances. (Incomplete efforts to privatize this system started in 2005 but the government still owns a majority of the postal banking system.)
Western Union in the United States was an attempt to imitate a postal banking system in the private sector, but it has never really gained a large market share outside the niche of international remittance payments to expatriates homelands.
None of these systems, however, is truly loan free or risk free, although they invest the savings of their customers in very low risk investments which are sometimes guaranteed by the government. Money in a modern economy is predominantly just negotiable third-party debt.
At the macroeconomic level of an entire economy, savings accounts and investments in real world economic assets are two inextricably tied sides of the same coin, however indirectly. Savings are a right to something of value in the future, but at any given moment in time, all real world assets have to be in use by someone. Real world goods and services can't miraculously time travel to the future from the time that you want to save money for future purposes. All of the goods and services available in the economy at any given time have to exist or be replaced by substitute real world goods and services at every moment of time between the time when you forego consumption by saving and the time when you want to withdraw your savings to consume real world goods and services.
In American popular culture, the classic illustration of the concept that savings are ultimately impossible to segregate from real world investments is a speech by George in the classic 1946 movie "It's A Wonderful Life" (paraphrased here), in which he explains to people in the midst of a run on a bank how their money isn't sitting in the safe but is instead tied up in mortgages on houses and the like. The key portion of the script for this purpose says:
MISS ANDREWS. But I’ve got my money here, George, and I’ll take it
now.
GEORGE. No. Miss Andrews, you’re thinking of this place all wrong. As
if I had the money back in the safe. The money’s not here. Your
money’s in the Grange’s house right next to yours. And in the Kennedy
house, and Mrs. Macklin’s house, and a hundred others. Why, you’re
lending them the money to build, and then, they’re going to pay it
back to you as best they can. Now what are you going to do? Foreclose
on them?
In the case of commodities (like gold) negotiable warehouse receipts are used in a similar way to a safe deposit box.
In the case of securities, a "nominee" holder often serves a safe deposit box like role. There is also a little known government agency that protects customers of these firms from events that deprive them of their securities (like embezzlement or destruction of stock certificates), in a way similar to the way that the FDIC protects commercial bank depositors.
An account in a Federal Deposit Insurance Corporation (FDIC) insured commercial bank up to the insured dollar amount is functionally equivalent although the "back office" way that it gets to functional equivalency is not the situation that you contemplate. Consumer authorized electronic fund transfers from commercial bank accounts insured by the FDIC, while they have had a legal framework in place for decades, have only very recently been used on a widespread basis by ordinary banking industry customers.
The existence of equivalents (from the small customer's perspective) like the FDIC insured banks is the operational reason that this isn't done, even though the macroeconomic link between savings and investments is the fundamental reason that this is the case. Borrowing cheap and lending high is a large share of the profit of a commercial bank that subsidizes its ordinary customer deposit account operations. Withdrawing this revenue stream would make it economically non-viable in competition with firms that do so.
