The idea that fractional reserve banking is fraudulent comes partly, I think, from confusion about different senses of the word “money.” Money in the narrow sense is the monetary base, which, at least from the standpoint of the domestic monetary system, is a pure asset and not somebody's IOU. Payment with the monetary base extinguishes IOUs.
(Asides: From the standpoint of the international monetary system, the monetary base may not be a pure asset if it is convertible at a set rate into something else, whether a commodity or a foreign currency, but ultimately, in all the cases I can think of, the trail ultimately leads to a pure asset. A government-issued pure fiat money is not an IOU because, though the government typically accepts it in payment of taxes, there is no fixed relationship between the fiat money unit and the tax paid or with any good.)
“Money” in the looser, broader sense includes IOUs, particularly those issued by banks, that are readily convertible at 1:1 into the monetary base.
A 100% reserve bank would be a kind of warehouse. Many banks offer warehousing services: they are called safe deposit boxes. I am aware of no bank for which safe deposit boxes are anything more than a side line of business, though. Because 100% reserves preclude lending funds out at interest, a 100% reserve bank would have to charge storage fees just as banks do for safe deposit boxes. Customers’ acceptance of interest paid by banks, or of “free” services offered by banks in lieu of interest, is an implicit recognition that customers are not warehousing assets, but allowing banks to act as their agents in lending out funds.
When we talk about “money in the bank,” we are talking not about the monetary base but about bank IOUs. It would be more accurate to use the phrase “credit in the bank,” because that’s what most of it is: IOUs by bank borrowers to which bank depositors are claimants.
The use of the word “money” to cover both things that are pure assets and those that are IOUs leads to confusion in other aspects of monetary economics also, especially when defining and discussing the money supply. That is a topic for another day.
ADDENDUM: Thank you for the comments, some of which raise issues I will address in the future (particularly Mike Sproul's point about the balance sheets of fiat money issuers). By a "pure asset" I mean something that is not a claim to something else. A lump of gold is a pure asset. So is perhaps the starkest example of fiat money: occupation currency, where an occupying army comes into a country, prints up notes, and forces people at gunpoint to accept the notes in payment for goods. Typically the notes are not freely accepted for payments to the home country of the occupying army. (I didn't say they were a good asset.) A "pure asset" is distinguished from an asset that is a claim to something else, such as a bond, which is a claim to a stream of future payments.