  For a quadratic field K, we investigate continuous mod p representations of
the absolute Galois groups of K that are unramified away from p and infinity.
We prove that for certain pairs (K,p), there are no such irreducible
representations. We also list some imaginary quadratic fields for which such
irreducible representations exist. As an application, we look at elliptic
curves with good reduction away from 2 over quadratic fields.
