Learn Russian: И умный студент, и опытный врач найдут выход в любой момент. - Both a smart student and an experienced doctor will find a way out at any moment.

любой means "absolutely any (no matter which one)", emphasizing that it doesn't matter which moment you choose. всякий tends to mean "every" or "any sort of" (referring to variety). For expressions of time where the meaning is "pick any moment at random", любой is the standard choice.

Why is there a comma before the second и in И умный студент, и опытный врач?

The repeated conjunction и ... и ... is the Russian way to say "both ... and ...". A strict rule in Russian punctuation is that when you use a repeating conjunction to connect items in a list, you must put a comma before the second one, and any subsequent ones.

Why is в любой момент in the accusative case here rather than the prepositional?

When talking about a specific point in time using the preposition в with words like момент, секунда (second), or days of the week, Russian requires the accusative case. Since момент is a masculine inanimate noun, its accusative form looks exactly like the nominative.

Can выход be used for both a physical exit and a metaphorical "solution"?

Yes! Just like "way out" in English, выход is commonly used in both senses. You will see it on signs above doors, but you can also use it to mean finding a solution to a problem, exactly as it is used in this sentence.