Edit: If you want realism, let us only carry what we actually could, meaning not three battle axes, 5 deadric cuirasses and 1000 arrows...
Banknotes were first used on earth that we know of at 700 AD.
Daggerfall had banks and banknotes.
There is nothing unrealistic about it.
I think the perfect solution would be money having weight, and be divided in gold, silver and copper coinage.
Each weighing 0.1 encumberance.
Banks. And banknotes that are like the old style promissory notes of yore, able to hold any denomination.
Anything but Oblivion style please, where gold was really just a stat.