There's a few things that you should be careful not to confuse here :
- If the universe obeys laws, this isn't the same thing as the universe processing those laws in a computer-like fashion to produce itself.
- The laws that we ascribe to the universe aren't the same thing as what is actually happening or what actually exists, if such concepts even make sense here.
- What exists and what we can measure are two different matters.
As far as we know, the universe has about the computational ability of a finite state machine. In other words, if you give a reasonably decent algorithmic system with finite memory (and a finite number of symbols), this is what our universe can do. The universe isn't even as powerful as a Turing machine, much less anything more powerful than that, although it certainly has a lot of possible memory compared to most computers.
Now here's a few arguments regarding what you want :
Measurements and systems are discrete
Any system that we'd care to measure (ie whatever we would like to use for memory), as well as any measuring apparatus we want to use on them, will give us discrete results. From quantum mechanics, if we have some system we would like to use as memory, represented by a wavefunction $\psi$, and an apparatus to look at the value stored in memory, represented by the wavefunction $\phi$, the measurement process goes thusly :
\psi \otimes \phi \to \psi_A \otimes \phi_A
The system is measured as being in the state $\psi_A$, (for instance its position or whatever else), while the apparatus is in state $\phi_A$ (ie, it's in the state where it indicates that the system measured is in state A). Those states are both eigenstates, and by the rules of quantum mechanics, both of them must be discrete (although their values could be arbitrarily high, but see later).
I could have used many other arguments as to why measurements are all discrete, but this is about as general as you can get, without getting bogged down in how the measurement is performed.
Measurements are finite
Obviously measurements made by humans have to be finite. We cannot ourselves observe an infinite length, or build a machine that displays an infinite number of digits, or distinguish two lengths that differ by an infinitesimal amount. But could we say, build a machine that can store arbitrarily high numbers in a state as memory, and in the end give us a finite answer? This would in effect give us infinite accessible memory.
The important part isn't the number of possible states in a system, but the fact that we can switch between any of those states (to change the memory, so to speak). Any change between two states will require some energy to be invested, energy which, due to entropy-related reasons, cannot be reused again. Therefore, if we only have access to finite usable energy (and we do), we cannot change such states to an arbitrary number of different states.
A finite playground
For quite a lot of reasons, but ultimately due to the expansion of the universe, we only have access to so many particles (everything else, if it exists, is beyond the cosmological horizon, which we cannot go into). This is a thing that restricts both our access to energy for changing states and simply particles we could use to do this. If the universe was static, then perhaps we could attempt such a thing (if you don't mind waiting trillions of years), but that is yet another reason we cannot.
What's going on with the universe
So far everything is about why we can't measure those things, rather than how the universe does it, but beyond the fact that the universe isn't a simulation, here's another reason why you should think hard about this.
The fact that the laws of nature are continuous has little to do with either measurements or a reasonable assumption. Physics mostly uses continuity because it is simpler to handle mathematically. Whether that reflects something more fundamental or not is another question.
It is always possible, both in principle and so far in practice, to replace continuous laws by discrete ones. The fundamental reason behind this is that we always have a finite number of measurements we base everything on. Mathematically, there is always a method by which to construct a theory that doesn't rely on anything continuous from this. More specifically for physics, it is entirely possible to make a theory where you get measurements in, calculate, and then get measurements out, without ever refering to anything continuous.
Perhaps there are objects which are continuous in the universe. Maybe spacetime is, for instance, but if that is so, we would have no way of extracting any such informations.