This is a special case of a theorem of A. Pfister. It is well known to quadratic forms specialists. See e.g. Theorem XI.2.6 in T.Y. Lam's Introduction to Quadratic Forms over Fields.
I believe the original paper is
Zur Darstellung von $-1$ als Summe von Quadraten in einem Körper. (German)
J. London Math. Soc. 40 1965 159--165.
In this same paper Pfister defines the "stufe" (which T.Y. Lam has successfully campaigned to be called the "level") of a non-formally real field, namely the least positive integer $n$ such that $-1$ is a sum of $n$ squares. Among his other achievements, he proves that the level is always a power of $2$ (so that Kevin Buzzard's recollection is correct). It is also worth remarking that his work is an insightful and rapid response to previous work of J.W.S. Cassels.