We construct an effective field theory of a two-neutron halo nucleus in the limit where the two-neutron separation energy B and the neutron-neutron two-body virtual energy ε_{n} are smaller than any other energy scale in the problem, but the scattering between the core and a single neutron is not fine-tuned, and the Efimov effect does not operate. The theory has one dimensionless coupling which formally runs to a Landau pole in the ultraviolet. We show that many properties of the system are universal in the double fine-tuning limit. The ratio of the mean-square matter radius and charge radius is found to be ⟨r_{m}^{2}⟩/⟨r_{c}^{2}⟩=Af(ε_{n}/B), where A is the mass number of the core and f is a function of the ratio ε_{n}/B which we find explicitly. In particular, when B≫ε_{n}, ⟨r_{m}^{2}⟩/⟨r_{c}^{2}⟩=2/3A. The shape of the E1 dipole strength function also depends only on the ratio ε_{n}/B and is derived in explicit analytic form. We estimate that for the ^{22}C nucleus higher-order corrections to our theory are of the order of 20% or less if the two-neutron separation energy is less than 100 keV and the s-wave scattering length between a neutron and a ^{20}C nucleus is less than 2.8 fm.