What is the smallest possible average of four distinct positive even integers?
The smallest average will occur when the numbers are as small as possible.  The four smallest distinct positive even integers are 2, 4, 6, and 8 and their average is $\boxed{5}$.

$\textbf{Note:}$ These numbers form an arithmetic sequence.  The average of the numbers in any arithmetic sequence is the average of the first and last terms.