Most capacitors are in the µF, nF and pF range. I know there are some special ones that go that high, but at the time faraday was around, and the unit was named after him, they didn't have such a thing. Why is the unit so large if we rarely use caps with that high of a value?
Electronic – Why is the unit for capacitance so large
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I would say that these are indeed non-polarized caps. You would want to replace them with similar capacitors. Do NOT replace them with ceramic caps.
Ceramic caps are great for many purposes but not in the audio path. Among other problems, they suffer from piezo electric sensitivity and their value can vary greatly depending on the magnitude of the DC voltage across them.
Digikey and Mouser have suitable replacements available.
Within reason, too large is not a problem. Also usually this kind of regulator will actually work fine with no capacitors (advertised as such on some datasheets) under some conditions, but I don't recommend it (maybe if you're just powering an LED). Other parts (eg. LM7905, many LDOs) are about guaranteed to oscillate without the proper output cap. Most commercial designs will use a somewhat higher values than the datasheet example (not really 'recommended') because there is no cost penalty and 0.33uF is not very common. I don't think I've ever seen the 330nF outside of hobby/internet hacks. Similarly 1uF is perhaps more common on the output these days- there is little cost penalty and it's a bit better. On the other hand, these parts are mostly used at relatively low current these days because switching parts are better suited once you get into the hundreds of mA or higher, so 0.1uF has as much effect as 1uF at much higher current.
Too big a capacitor on the output might affect startup because the regulator can only supply ~1.5A to charge the capacitor, otherwise larger is somewhat better all other things being equal. If the ESR is larger a large output cap may not be that much help. Low ESR is not a problem. A 1uF ceramic (or 0.1uF) works well and is almost free. Here's a plot of output Z vs frequency- at a few hundred kHz the output cap starts to have an effect. Below that, no capacitor is typically as good!
The 3300uF is probably there as a mains filter, not anything to do with the regulator directly. If the input voltage drops too low the regulator cannot regulate so the input reservoir capacitor has to be adequate for the lowest input mains voltage, highest load and end of life capacitance value. The input ripple also gets passed to the output to some degree (-54 to -80dB - see datasheet). For the regulator's stability purposes, a small ceramic or electrolytic capacitor close to the regulator (within a couple cm) is fine. A 47uF/50V 105°C electrolytic for through hole or another 1uF ceramic works fine. If the input voltage gets too close to the point where it drops out then performance will tend to suffer even before it drops out of regulation- 3V is good.
Low ESR on the output caps does not make the regulator more immune to HF oscillation. It will generally improve the transient response to high frequency load changes. It's not unusual to have several uF of low-ESR ceramic caps distributed around a board near various chips. Several dollars? $2.61 will buy 1,000 0.1uF 10V X5R capacitors at Digikey if you don't mind compact 0402 parts.
Yes, both some negative and most LDO regulators are sensitive to ESR range- too high or too low and some will or may oscillate under some conditions. The LM780x is not an LDO regulator- it's actually quite difficult to convince it to oscillate. The LM790x, despite the similar name and the fact it's not an LDO is very easy to make oscillate (mostly because the output is taken from the collector of an NPN transistor rather than the emitter of a PNP as it would be if it was a true complementary part).
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Best Answer
As others have mentioned, 1 farad is 1 coulomb per 1 volt. But the rabbit hole goes deeper -- the question then becomes why is 1 coulomb what it is, and why is 1 volt what it is?
Following this rabbit hole to the bottom will eventually lead us to the 7 base SI units, which are units of measure for the 7 physical attributes of our world: distance, mass, time, electric current, thermodynamic temperature, amount of a substance, and luminar intensity. They're like axioms in mathematics. From here, other units are defined in terms of these. So volt = (kilogram meter meter) / (ampere second second second). Meanwhile coulomb = ampere * second. You'll notice that 1 of a derived unit is expressed in terms of 1's of a base units.
So ultimately, 1 farad is so large because the base units are so large, at least relative to the sizes of electronic components nowadays where we fit billions of transistors onto several square millimeters.