I worked as a financial analyst inside of Kodak during the 1980's and 1990's. Many of the answers already provided are fairly accurate but I'd add that in hind sight it looks like a 'no-brainer' that was missed by "old men stuck in the 50's"-while in real time it wasn't so clear.
1) The company enjoyed a huge market share of the photographic film and paper market around the world. This was a source of most of the companies profits and cash flow. The entire economic engine was based on customers having to buy 'consumables' to enjoy their images and videos.
2) The biggest threat was believed to be Fuji - because they had as good if not better quality, and even more distressing to Kodak, at a lower cost.
3) I was in a strategic planning meeting that I distinctly recall that we were reviewing our SWOT analysis. During the "Threat" section discussion the threat from digital photography was addressed. The "moment" that was burned into my memory was the "executive level" input that said "digital photography will grow the consumption of photographic paper". This was seen as the most rational, lucid, and ironically, the most 'progressive' opinion inside of Kodak.
4) In hindsight, HP was the real threat, not Fuji. Once HP printers achieved a"good enough" resolution for photo prints, and at the same time digital cameras also became "good enough" at a low enough price point, the death of Kodak's cash machine was cast. It was just a matter of time.
5) What was the big miss? One was that consumers demanded '(silver halide) photo grade' resolution--that the bar for "good enough" was way higher than it turned out to be. As "good enough" became much 'lower' in terms of technical specs-so did the cost point. The rest is history.
From my perspective, there was a great deal of inertia which kept the status quo as the only legitimate strategy. As Kodak acquired non-silver halide businesses, especially high tech, digital businesses, there were numerous clashes of culture, decision making, speed of execution, and layers of management, etc.
However, I would be remiss if I didn't also say that Kodak was a great company, it had excellent processes (not just manufacturing, but human processes, software processes, management processes, etc.). It attracted and retained great people who were among the most dedicated people I have ever had the pleasure of working with and not everything the company did was the result of "old men stuck in the 50's", it was progressive in many many ways.
Which is to say it would be a wonderful research project, not just on strategic failure, but also on how to build cultures, how to empower people, etc.
I cherish my time with Kodak, its unfortunate history causes me to pause even now and ask "what are we missing, right now, that in 5-10 years we will say was a "no brainer" that we failed to grasp"?