We investigate close binary neutron stars in quasiequilibrium states in a general relativistic framework. We assume conformal flatness for the spatial metric and irrotational velocity field for the neutron stars. We adopt the polytropic equation of state. The computation is performed for the polytropic index n(=0.5, 0.66667, 0.8, 1, 1.25), and compactness of neutron stars M/R(=0.03 - 0.3). Results of this paper are as follows. (i) The sequences of the irrotational binary are always terminated at an innermost orbit where a cusp (inner Lagrange point)appears at the inner edges of the stellar surface. The binaries with cusps are found to be dynamically unstable for n=0.5 and stable for n > 0.8 irrespective of M/R < 0.2. For n=0.66667, the stability changes depending on M/R. (ii) The gravitational wave frequency at the innermost orbit turns out to be between 800 and 1500 Hz for realistic compactness 0.14 < M/R< 0.2. (iii) The ISCO for n=0.5 appears to be determined by a hydrodynamic instability for M/R < 0.2. We derive fitting formulae for the relation between the orbital angular velocity at the ISCO and the compactness to clarify it. (iv) The maximum density of neutron stars in binary systems slightly decreases with decreasing the orbital separation and hence they are stable against individual radial collapse during the inspiral. (v) q = J_tot/M_ADM^2 at the innermost orbits is always less than unity for M/R > 0.13 irrespective of n, which indicates that the realistic binary neutron stars satisfy a necessary condition (q<1) for formation of a black hole before the merger. (vi) The specific angular momentum of any mass element in irrotational binary neutron stars at the innermost orbit appears to be too small to form a disk around black holes formed after the merger.