Genuine, and the missing clause is the part that explains what he meant.
Quoted short, it sounds like a slogan about bravery — almost a taunt. Quoted whole, it is a diagnosis. Roosevelt is not telling a country in the depths of the Depression to cheer up; he is naming a specific psychological failure and describing what it does.
The three adjectives are doing the work: nameless, unreasoning, unjustified. He is distinguishing between rational fear of real conditions — millions unemployed, banks collapsing — and the formless dread that stops people acting at all. Only the second kind is what he says we have to fear.
And the ending matters most: terror that paralyzes needed efforts to convert retreat into advance. The problem with fear, in his account, is not that it feels bad. It is that it stops the work.