Learn Russian: Учёные годами безрезультатно искали ответ. - Scientists searched for an answer in vain for years.

It demonstrates the Instrumental of extended duration. When time nouns are put in the bare Instrumental plural, like годами, they show that an action went on continuously or repetitively for a long, grueling time. It translates well to "for years" or "for years on end."

How does годами feel different from just saying много лет (many years)?

Много лет is a neutral, factual statement of duration. Годами adds emotional weight, emphasizing the prolonged, uninterrupted nature of the search. It highlights the process of time dragging on, fitting perfectly with the frustration of doing something безрезультатно (in vain).

Why does the noun учёные (scientists) have an adjective ending?

Учёный is an adjectival noun. Originally an adjective meaning "learned," it functions entirely as a noun meaning "scientist" or "scholar" but continues to decline exactly like a regular adjective. Here, it is in the nominative plural.

I've learned that искать (to search) can take the Genitive case. Why is ответ in the Accusative here?

The verb искать tends to take the Genitive when you are searching for something undefined or highly abstract, like seeking sympathy or a general sense of truth. However, when looking for a definitive, expected item, like "the answer" to a specific scientific problem, Russian prefers the Accusative case (ответ).