If the Saturn V rocket along with its Apollo spacecraft was miniaturized, for example to 1/72 scale so it was five feet tall, could it still perform a moon landing like the Apollo missions and get back to Earth? The rocket equation only involves the percent mass of the rocket as propellant for a certain delta-v, so it seems that a miniature Saturn V would have the same total delta-v as the real thing.
If you could miniaturize each component uniformly, you're correct that the rocket equation terms would all balance out and the rocket would be capable of the same delta-v performance.
However, the impact of atmospheric drag would be much worse; drag force is proportional to cross sectional area, not to volume. A Saturn V loses less than 1% of its delta v potential to drag; a five foot version would probably lose so much to drag that it couldn't reach orbit.
I imagine there would be other square-versus-cube-scaling problems involved in ridiculously tiny fuel pumps, injectors, and combustion chambers as well, probably making the engines vastly less efficient or preventing them from working altogether, but since I don't know much about those issues we will assume some sort of elfin magic is taking care of that end of things.
You've got a few problems:
When a model is scaled to 1:2 it's size, its area drops by 1/4 but its volume by 1/8. This is known as the square-cube scaling problem. So you will have one quarter the drag but one eighth the impulse. Russell expands on this, so I'll address the other issues.
The Reynolds number of the air does not scale with the model. So you not only have the square-vs-cube scaling problem (which could be mitigated overcome by making the model a bit longer), you've got probably an order of magnitude more drag per unit of area. You can think of this as the air being thicker from the craft's point of view. At extreme (nano) scaling the air molecules can even act as discrete particles colliding with the model!
If acceleration is scaled then the model will spend much more time in the atmosphere. Each second not in orbit will use up to 9.81 m/s of delta-v on gravity drag, no matter how much your craft weighs or how much propellant you brought with you.
In addition to the other answers, you'd also have to take temperature into the equation.
Your cryogenics tanks have much thinner thermal insulation, primarily LH2 (LOX was "insulated" by the tank hull + ice, yes, really) but the cryogenic is still at the same low temperature. Much more ice build-up on the outside in relation to the Saturn's total weight. Much more pressure build up in the tanks due to increased boiling. Instead of a smooth lift-off I'd expect some crackling, rupturing and hydrogen/oxygen rivers on the pad...
Even setting aside the other problems, rocket engines themselves do not scale indefinitely.
You decrease the scale of a rocket motor, you decrease the thickness of the rocket chamber walls, which in turn decreases the bursting pressure of said rocket chamber.
If you scale the rocket down enough, this will become a limiting factor.
At that point you will have to drop the pressure of the rocket chamber to avoid bursting it, which, since the maximum specific impulse achievable by a chemical rocket is limited by (among other things) chamber pressure, means that there will come a point where your specific impulse has to drop as you decrease the scale.
And sooner or later (read: sooner), it will reach a point where the specific impulse is low enough that the effective delta-v is too low to reach orbit.
You have analogous problems with thrust (pressure limited, again), temperature inside the rocket chamber (surface area to volume ratio increases as the scale decreases, causing the percentage heat loss to rise), and tankage (pressure limited, yet again).
Also don't forget that the tanks would be thinner in general, making it much more likely to break through the tank, and suffer problems. The tanks need to be able to withstand the pressure and the weight of the payload and rocket. Thin walls would probably still hold the weight, but they probably wouldn't hold the pressure.
Overall, it'd be nice if this would work, but there are a lot of problems in terms of miniaturization, and all orbital rockets are big for a reason.