When the plates are really close, the electric field between them will get really high ( E = V/d, where V is the voltage difference between the plates and d the separation distance). If this electric field is very high, the air between the plate will become a conductor (dielectric breakdown) and thus the capacitor will be discharged. Generally, we put a plastic material between the plates to increase the breakdown limit (dielectric screening), but if the distance between the plate is really small, it will be impossible to negate the breakdown effect (because nothing will fit to screen the electric field).
It's also worth pointing out that no matter how amazing a dielectric you use, once it gets thinner than a couple nanometers electrons will just blow through it through quantum tunnelling.
EDIT: It seems there's a bit of a discussion brewing about the relevance of this to modern computer chips and specifically the MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) that is the work-horse of modern computers and microchips.
Maybe I'll just make some comments up here on this. A MOSFET is so-called because you have a metal "gate" (the M) which is separated from a semiconductor channel (the S) by a thin layer of insulating material. Specifically, the insulating material is generally silicon dioxide (thus, the O), because it is basically "silicon rust" and thus it's easy to grow such a layer on top of silicon, just heat its surface in the presence of oxygen.
A MOSFET transistor is basically a "valve" where if there are electron sitting at the gate then charge can freely move through the channel (the valve is on), if there are not, then the channel is basically an insulator. Thus you have a switch (0 or 1). This all works by the capacitive effect of the electric field felt in the channel DUE to the electrons on the gate. If there is an electric field in the channel the material in the channel is conductive and electrons can pass, no E-field and it's an insulator.
As we try and fit more MOSFET transistors per silicon chip, each MOSFET must get smaller and generally the oxide layer must get thinner. However, the key to operation is that electrons on the gate need to stay on the gate and only influence the channel through their electric field. However, with thinning carriers can start to tunnel from the gate to the channel (in addition to other mechanisms, like hot-carrier injection).
This is not some heady-new issue at the fringe of technology, this has been a fact-of-life for device designers since at least the 90s. So this inability to thin a capacitor beyond a certain point because of quantum tunnelling is a big reliability issue in how computers work.
Though it's also worth point out that sometimes tunnelling through the oxide is the whole point. Flash memory and SSD memory, something that is a ubiquitous technology today, WORKS through quantum tunnelling through an oxide. Specifically, the basic element of such memory is, what is called, a Floating-Gate MOSFET (FG-MOSFET). Which works very similarly to a regular MOSFET except between the gate and the channel, in the middle of the oxide, there's an "island" or "floating gate" that is totally insulated from everything else. A 1 or a 0 in such memories depends on whether there is charge on this floating gate. Thus, even if the device is shut off, those electrons on the floating-gate stay there (ideally) and make the switch "always on" by their electric field if the island is populated. You can then read a 0 or 1 by seeing if the "valve" is open or closed.
But how do you GET the electrons on this insulated island? Quantum tunnelling usually (some older devices use a mixture of quantum tunneling and hot-carrier injection).
As the plates get closer together, the electric field gets stronger, so it gets harder to prevent charges from jumping across the gap and shorting out the capacitor, and harder to hold the plates a precise distance apart.
However, these problems can be overcome, and there *are* very small 1 farad capacitors. The're about the size of a peppermint, and cost [a few bucks.](https://www.mouser.com/Passive-Components/Capacitors/Supercapacitors-Ultracapacitors/_/N-5x76s?P=1yztzayZ1z0wug0
) If you're used to dealing with nano- and microfarads, they're pretty miraculous!
They work by replacing one of the plates with a liquid containing salt ions. The ions are attracted to the charged plate, but the liquid molecules keep the ions from actually touching it, so the gap between the two "plates" is just a couple molecules thick.
But for the reasons mentioned earlier, these supercapacitors have very low voltage limits (usually around 5 volts max). They also respond slowly to voltage changes, because the ions have to move around in the liquid.
Capacitance is indeed inversely proportional to the distance between plates, but it is also proportional to the overlap area of the plates. You can theoretically increase capacitance by bringing the plates closer together, but if you're shrinking the plates at the same time, you may not gain much.
Another view: The energy stored in a capacitor is **½CU²**.
The volume is the plate area **A** times the distance **d**.
If you half the distance, you double the capacitance. However the voltage limit as well as the volume is now also ½(approx). As you now can not charge the capacitor as much you get double the capacitance times ½ the voltage squared, so the total energy is half that of the original capacitor. As the volume decreased as well, the overall energy per volume stays constant.
Conclusion: in order to store more energy in a capacitor, you need to change the dielectric, not the plate distance.
The engineering challenge is decreasing the distance as much as possible, while preventing discharge. If you put too much voltage on the "plates" at too close of a distance, then just like when a spark is generated, electric breakdown occurs.
What to have smaller voltage? Fine, you can pack the plates super close and have tiny capacitors. Need a higher voltage rating? Darn, need to separate the plates to have higher voltage standoff.
Beyond that, it's really a question of application. If you need capacitors with high energy capacity, higher voltage is your best bet. Need one with better high frequency response, then need to use certain materials.
There is no magic capacitor that can do everything. For each application you exploit one or two critical traits but there are trade offs for that.
Integrated circuit chips make use of capacitors on the order of nanometer size, however their capacitance is nowhere near the level of Farads. More on the order of femto-farads (10^-15 F)
It's also noteworthy that they typically make use of a high-K dielectric materials such as Hafnium nitrido-silicate.
Such capacitors have extremely high leakage current due to the small gap. So they're only typically useful for high frequency switching applications like IC's. The self-discharge time is very small.
If you can find a thin material that insulates extremely well to wedge between the plates, thats exactly what you will have. The complication arises with maintaining a gap current can't pass through but an electric field can.
The shorter the distance, the lower the breakdown voltage - it's generally (at least theoretically) proportional to the distance.
Using different materials can increase the dielectric breakdown voltage, but in any case that breakdown voltage is typically measured in volts per some unit measurement. E.g. air is typically around 10,000 volts per inch (at least to a rough approximation - of course changing temperature, pressure, or humidity of air will alter that, as will altering composition - e.g. ratio of gasses, type and density of contaminants (e.g. dust), etc.).