Unfortunately, much of our universe appears to be quantized. One could imagine a system wherein, instead of rolling dice, you precisely measured the net magnetic spin of some ideal gas on non-interacting magnetic moments each of either spin up (1) or spin down (-1). Theoretically such a property, when measured many times, should form a gaussian distribution (a bell curve) with an exceedingly small standard deviation. Unfortunately, this is not quite right. In actuality only certain discrete values of net spin are possible, based on the number of moments in the system. For example, if there is an odd number of moments, it is impossible for the net spin to be exactly 0. Also, obviously, it is impossible for the net spin to ever be any number that is not an integer or to be larger than the total number of particles. All of these concerns have a negligible impact on the behavior of large collections of particles, but they mean that, at a fundamental level, you don't have a continuous distribution.
The example deals with a very idealized situation, but really a great many physical quantities have been shown to be (very probably) quantized by modern physics. The question, then, requires us to find something that isn't.
Oddly, position is currently not believed to be precisely quantized, though this is disputed by some physicists. In the standard model of quantum mechanics, when one puts a single electron in the ground state in an infinite square well, a truly continuous probability distribution (the Schrodinger Equation) describes the likelihood thereafter of finding the electron to be at any given point in space within the box. Thus, with some sophisticated and expensive experimental physics equipment, one could actually maybe be measuring a continuous probability distribution. But here we encounter the second, and more fundamental issue with your plan:
The Arabic numeral system is inherently discrete. Most measuring devices give their outputs in numbers (citation needed). Lets take the number 5.5563, for example. If this is the number our instrument reports, the next possible number is 5.5564. The instrument can't report 5.55631, or any other number between the two possible results. In order to have a continuous distribution we would need an infinite degree of precision and an infinite number of digits. This is a problem.
Clearly the solution is to use a measuring device that doesn't report numbers. Unfortunately, unless it its measurement is based off of gravitational interaction, the act of measurement itself now creates problems for us. All force interactions other than gravity possess a good amount of evidence supporting the existence of a mediating particle. Such mediating particles result in quantization of the interaction, so that only interactions involving an integer number of such mediating particles (quanta) are possible. While people like to speculate about gravitons, there really isn't much evidence yet for their existence and a far bit of evidence to the contrary, so you're actually on pretty solid ground if the measurement device is solely using gravity to measure the position of the electron in the box and reporting that position by means of deflection in its own position (the uncertainty in the latter is of no concern to us; indeed it is helpful to you).
So now we have a continuous probability distribution measured and reported in a continuous manner. So we're good right? Not quite.
Gravity is very weak, and electrons are very small. Unfortunately, large (i.e. massive) objects have an uncertainty in position that is astronomically small. So yes, this works, but you can't see it, which kinda defeats the purpose of a measuring device in the first place. So, for all intents and purposes, it is not currently technologically feasible to generate a random number from a truly continuous probability distribution. In order to do so we'd need something like a macroscopic atomic (i.e. indivisible) object with the mass of an electron, or some such.
And even then quantization would interfere if you were telling where it was by looking at it, feeling it, smelling it, or otherwise measuring it in a physical way via biology.
So, you're pretty much screwed.
But wait! There's hope! The human mind is a powerful thing, and evidence suggests it can create and emulate a continuous probability distribution. You can experience the spiritual and mental directly, rather than indirectly as with the physical, and you can control directly the physical and mental as you cannot the spiritual. So then, if you were to conceptualize a continuous probability distribution and by virtue of the free will granted to you by God you selected a truly random point in the distribution you could have a number of the data type you need to make this work. Do any systems currently existing not only have you do that but also ask you to do math on it? No. Not really.
There are systems that ask you to do this, though. In Amber Diceless, the GM is supposed to (if I understand it correctly) come up what exact thing is retrieved by quick Logrus summoning via a random result with a normal distribution centered about the desired item and the standard deviation directly proportionate to the reciprocal of the time use to search for the thing. This is the only example I am aware of.