I was watching Roman Empire on Netflix. In season 1 episode 2 (minute 14:41) it's stated that in 37 AD Caligula fell seriously ill. His illness is called "cerebral fever" and the documentary portrays it as a comatose state. It's also said that it took Caligula three months to wake up and finally recover.
How did he survive three months in a coma at a time when modern artificial nutrition techniques were not available?
He didn't. The coma was a conjecture made by A. T. Sandison in 1958, and I don't know where the three months originated but that's conjectural too. Here's a response on the same question that I wrote a few months ago (very slightly edited):
When a story like this seems so bizarre, it may be that something is being misreported.
And it is misreported. No ancient source mentions a coma. No ancient source mentions a period of three months. No ancient source mentions headaches.
The sources for Gaius' illness are Philo, Embassy to Gaius 14-21 (ca. 40 CE); Suetonius, Gaius 14.2 (ca. 100 CE); and Dio Cassius 59.8.1-2 (3rd century).
It's clear the illness was severe, and perhaps life-threatening, and that it lasted long enough for news to propagate around the empire, that is to say on the order of a month, perhaps more. But beyond that, what they describe is vague. There are no descriptions of symptoms.
Philo. In the 8th month of Gaius' reign, in early autumn, a serious disease struck him. (The 8th month of his reign ought to have been October-November 37, mid-to-late autumn, so Philo's timeline is a bit wobbly.) The illness arose because he had changed his behaviour from relative modesty, while Tiberius was alive, into a lifestyle of sumptuousness and gluttony. While he was ill people put on a show of mournful faces, and when news came of his illness abating, they were on tenterhooks for the latest news. When it was announced he was fully recovered, they went back to their normal cheerfulness.
Suetonius. A few months after the beginning of his reign, Gaius fell ill, and courtiers put on a big show of making ridiculous vows if his life should be spared.
Dio Cassius. Gaius became sick, but didn't die, and instead caused someone else's death -- Tiberius' grandson, on suspicion of trying to profit from Gaius' illness.
Suetonius and Dio aren't great sources, but Philo actually met Gaius and was writing before the end of his reign. Together they make a reasonably coherent story (maybe setting aside Dio's lurid story about Tiberius' grandson).
For a take on the evidence and what people have thought about it, I suggest Anthony Barrett's biography Caligula (1989, repr. 2001), the opening pages of chapter 5. He sums up some 20th century speculations about the nature of the disease: encephalitis lethargica, a.k.a. sleeping sickness (Sandison, 1958); hyperthyroidism (Katz, 1972); something affecting the central nervous system (Massaro and Montgomery, 1978); or a particular form of epilepsy (Benediktson, 1989). You may also have read an earlier version of these conjectures in Robert Graves' novel I, Claudius (1934), which also casts the illness as transformative, causing delusional insanity.
These conjectures are based mostly on the supposition that there was a change in Gaius' character before and after the illness -- Suetonius and Dio seem to regard it as a turning point in his career, and Philo draws a causal link to Gaius' lifestyle -- in conjunction with Suetonius' testimony that as a young man Gaius suffered from "falling sickness" (comitialis morbus, Gaius 50), and sometimes had faint spells where he was unable to walk or collect his thoughts. The falling sickness in his youth is worth taking seriously, but we don't know if it was connected to the illness in 37, and the notion of a change in Gaius' character is barely more than speculation.
Sandison's suggestion of encephalitis lethargica (article link; a.k.a. epidemic encephalitis, a.k.a. sleeping sickness) seems to be the one that has taken hold in popular accounts. That's probably because it's the only one that appeared in a medical journal (obviously historians can't be trusted!). That isn't to say it's incompetent: Sandison shows mastery of the ancient sources, of how unreliable they are, and of the secondary literature.
Anyway Sandison's guess is, I reckon, where the notion of a coma came from. You'll find it widely reported that Gaius suffered from headaches before going into his supposed coma: as I mentioned, that isn't mentioned in any ancient source. It must be entirely based on the supposition that he had encephalitis.
If you compare the specificity of the suggestion with the vagueness of the sources, the only responsible conclusion is that we don't have anywhere remotely near enough information to infer any symptoms at all, let alone to diagnose a very rare illness.