A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.) Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.