Algebraic K-theory space, as a motivic space, has a known geometric model given by (Z copies of) the infinite Grassmannian. In the new geometric model that we offer, the infinite Grassmannian is replaced by the Hilbert scheme of the infinite affine space, stabilized with respect to degree. Although the geometry of the Hilbert scheme is much more complicated, this model leads to new insights on K-theory. For example, we obtain an analogous model for the hermitian K-theory space, given by the Hilbert scheme of finite Gorenstein subschemes that are equipped with an orientation. Turns out, this model works also in charactersitic 2, while the previously known model, given by the infinite orthogonal Grassmannian, represents hermitian K-theory only over a base where 2 is invertible. This is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro.