To find the elements of \((X \cap Y)'\), we first need to determine \(X \cap Y\) and then find its complement with respect to the universal set, which includes all elements in \(X\) and \(Y\).

1. Find \(X \cap Y\):
- \(X = \{1, 3, 5, 7, 8, 9\}\)
- \(Y = \{2, 4, 6, 8, 10\}\)
- The intersection \(X \cap Y\) is the set of elements common to both \(X\) and \(Y\).
- \(X \cap Y = \{8\}\)

2. Find \((X \cap Y)'\):
- The universal set \(U\) is the union of \(X\) and \(Y\).
- \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
- The complement of \(X \cap Y\) with respect to \(U\) is all elements in \(U\) that are not in \(X \cap Y\).

So,
\[
(X \cap Y)' = U - (X \cap Y) = \{1, 2, 3, 4, 5, 6, 7, 9, 10\}
\]

The correct answer is C. \{1, 2, 3, 4, 5, 6, 7, 9, 10\}.