From the conversation of the sailors, we noted the following detail this year:

Cook: When I was as old as the sailor is now, I was twice as old as he is.

Engineer: I am only 4 years older than the sailor.

Sailor: Only the cook's age is an odd number, and the least common multiple of our three ages is the captain's birth year.

How old is the captain?
From the sailor's statement, it follows that his age is an even number. Let's denote it as $2x$. Using this, express the ages of the others. From the engineer's statement, we know that he is $2x+4$ years old. From the cook's statement, it follows that he is one and a half times as old as the sailor, that is, $3x$ years old. Therefore, the captain's birth year is the least common multiple of $2x$, $3x$, and $2(x+2)$. Here, $x$ represents an odd number because the cook's age is an odd number. Consequently, the numbers $2$, $x$, and $x+2$ are pairwise coprime.

Case I. If $x+2$ is divisible by 3, then the captain's birth year is $2x(x+2)$. Let's examine whether this is possible! The function $y=2x(x+2)$ is strictly increasing for positive $x$ values. If $x=25$, then $y=1350$. This is too early to be a birth year. The next possible value for $x$ is 31 (since $x$ is an odd number and $x+2$ is divisible by 3). But then $y=2046$. This is not a feasible birth year.

Case II. If $x+2$ is not divisible by 3, then the least common multiple is $z=6x(x+2)$. This is also an increasing function for positive $x$. For $x=17$, $z=1938$, which is a plausible birth year. If $x$ is smaller or larger than this, we again get unrealistic values. Therefore, the captain was born in 1938, making him 40 years old.