Most of the wealth was in assets, as you say. Crassus owned a lot of property, for instance. Almost all wealthy Romans possessed multiple structures, either in the city or in the form of rural estates. Many had a nice vacation house on the cliffs at Baiae, as well. In addition to property was livestock, agricultural produce, and the like.

Yes, there had to be some physical currency around. We have some evidence for the storage of valuables in houses, especially from Pompeii. The house of Obellius Firmus contains an arca, or strongbox, which might have been used to store bullion or coinage. Scroll down for pictures and a reconstruction drawing. Actually inside it was found a shovel, a lamp, and an old whetstone, none of which needed to be there. I agree with the excavators that these items probably did not normally hang out in the arca.

There were also public/civic coffers. We know that Rome stored physical "money" in the substructures of the Temple of Saturn in the Forum. When Caesar "crossed the Rubicon" in 49 BCE, he went straight for the temple (wars are expensive business). Pliny the Elder (33.56) writes about it a century later:

Julius Caesar, entering Rome for the first time after the beginning of his civil war, took from the Treasury 15,000 gold ingots, 30,000 silver ingots, and 30,000,000 sesterces in coin.

Plutarch (Caesar 35) elaborates:

Afterwards, when Metellus, the tribune, would have hindered him from taking money out of the public treasure, and adduced some laws against it, Caesar replied that arms and laws had each their own time; "If what I do displeases you, leave the place; war allows no free talking. When I have laid down my arms, and made peace, come back and make what speeches you please. And this," he added, "I tell you in diminution of my own just right, as indeed you and all others who have appeared against me and are now in my power may be treated as I please." Having said this to Metellus, he went to the doors of the treasury, and the keys being not to be found, sent for smiths to force them open. Metellus again making resistance and some encouraging him in it, Caesar, in a louder tone, told him he would put him to death if he gave him any further disturbance. "And this," said he, "you know, young man, is more disagreeable for me to say than to do." These words made Metellus withdraw for fear, and obtained speedy execution henceforth for all orders that Caesar gave for procuring necessaries for the war.

A few things. First, Crassus became rich by land speculation (similar today to how people can be worth a certain amount of money, but that money can be in a less liquid form like real estate, equities, securities, etc., rather than just cash). Specifically, he basically owned a private joint real estate, insurance, and fire fighting company that would go around looking for burning down buildings, buy them very cheaply from the owners, put out the fires, and refurbish the buildings to make a profit (Plutarch, Life of Crassus 2).

There were ways for rich individuals to store their money in the form of coinage. One way was to keep coin hoards, which were basically big pots filled with coins, which were then buried (basically buried treasure).

There were also rudimentary banks during the Republic, a system somewhat adopted from the Greeks. The Greeks would often use temples to mint coins and set up public depository systems. The Romans mostly privatized this system during the Republic as private individuals could mint coins and set up deposits for other people (as well as act as money lenders).

Banking during the Republic, though, was very rudimentary, but did develop into a more modern system during the Empire (though it did break down during the Crisis of the Third Century, not to be reestablished in Europe until the 11th or 12th centuries).

You can find more info on this in Andreau's Banking and Business in the Roman World.