 Solution

No such ring exists. Characteristic two gives $r+r=0$ for every $r$. Choose a nonzero element $a$, which exists in a five-element set. Translation $r\mapsto r+a$ is an involution without fixed points: applying it twice gives $r$, while $r+a=r$ would force $a=0$ by additive cancellation. Its disjoint pairs would partition all five elements, which is impossible because five is odd.

This argument applies even to a nonunital ring. With the unital convention, it is also the case $n=2$ of finite ring characteristic divides cardinality.