A partial answer:
If $k$ is an ordered (formally real) field then it should be a real closed field. Since every irreducible polynomial of degree $3$ gives rise a formally real extension of degree $3$. So $k$ is elementary equivalent to real numbers.
In the below paper it is proved that every element in the full matrix algebra $M_3(k)$ is a sum of two idempotents if and only if every polynomial of degree $3$ has a root in $k$.
de Seguins Pazzis, Clément; On decomposing any matrix as a linear combination of three idempotents. Linear Algebra Appl. 433 (2010), no. 4, 843–855.