My reasoning for this is as follows: you first pick two from 10, then two from 8, et cetera until nobody is left. The order doesn't matter, so you also divide by the number of permutations of the 5 groups:
$$
\frac{C_2^{10} \cdot C_2^8 \cdot C_2^6 \cdot C_2^4 \cdot C_2^2}{5!}
$$
I'm not at all confident with combinatorics though; is this formula correct?
Best Answer
Your answer is correct, but it may help also to see an answer that avoids "division by symmetry" arguments. Another way of expressing the answer is $9!!=9\cdot 7\cdot 5\cdot 3\cdot 1=945$ using double-factorial notation.
To arrive at this answer recognize that for our ten people, there must be some way of distinguishing them and therefore ordering them. I will continue this explanation as though we ordered them by height.
Approach by multiplication principle:
Pick who the person to be partnered with the shortest person is. (9 options) Remove both the shortest person and his chosen partner from the available people.
Pick who the person to be partnered with the shortest remaining person is. (7 options) Remove both the current shortest person and his chosen partner from the available people.
Repeat this process until all people have received partners.
This gives $9\cdot 7\cdot 5\cdot 3\cdot 1=945$ possibilities.