It turns out that Switzerland is smaller than the Los Angeles metro area both in GDP and population.
It seems it's comparable in size, but not smaller
Take a look http://mapfrappe.com/?show=40241 (it's a rough sketch of Switzerland but the size should be right)
Edit: yes, I think parent meant population, not actual size
Edit: Also, Wikipedia is possibly a better source for comparing sizes of landmasses.
LA for example is only about 66 percent: http://www.laalmanac.com/education/ed33a.htm
This comes up all the time, especially in "Why can't the US do X?" discussions but isn't it just the equivalent of shouting down with "That will never scale!"
Of course it might not be exactly the same. Maybe the bigger country will need more servers spun up or a tweaked algorithm (to continue the analogy), but just saying a country is smaller doesn't mean things are a No before even being tried. I'm assuming there were railroads across other smaller countries before the US had the transcontinental railroad too.
So, specifically in the context of discussing direct democracy, size does matter.
Even in the relatively recent founding history of the United States, the Federalist Papers discuss and try to address a variety of issues which can lead to this outcome.
And ultimately, I think this is what we see with the Brexit outcome and the current Trump campaign in the US. These aren't "new" concepts...throughout history we can see similar outcomes for similar events with similar reasons.
You raise an excellent point though, which is with the ease of communication in the modern era, why don't we have a more informed public? This is something I have thought about trying to tackle with a side dev project...but the issue is complex and difficult to address directly.
Side note, I believe the Roman Empire was a Republic, not a Direct Democracy :)
Nit pick: the Roman Empire, in one sense, was founded while Rome was still ruled by the Republic; in this sense, the Empire is the metropole-periphery arrangement with Rome proper ruling as the metropole and the various provinces as peripheries.
In the other sense (the sense of the Empire as a form of government within the Rome rather than the relation between Rome and the provinces), the Roman Empire was founded when Caesar Augustus was granted a variety of lifetime powers by the Senate.
In either sense, it was not founded by (or during the life of) Julius Caesar.