The gap structure of Sr$_2$RuO$_4$, which is a longstanding candidate for a chiral p-wave superconductor, has been investigated from the perspective of the dependence of its specific heat on magnetic field angles at temperatures as low as 0.06 K ($sim 0.04T_{rm c}$). Except near $H_{rm c2}$, its fourfold specific-heat oscillation under an in-plane rotating magnetic field is unlikely to change its sign down to the lowest temperature of 0.06 K. This feature is qualitatively different from nodal quasiparticle excitations of a quasi-two-dimensional superconductor possessing vertical lines of gap minima. The overall specific-heat behavior of Sr$_2$RuO$_4$ can be explained by Doppler-shifted quasiparticles around horizontal line nodes on the Fermi surface, whose in-plane Fermi velocity is highly anisotropic, along with the occurrence of the Pauli-paramagnetic effect. These findings, in particular, the presence of horizontal line nodes in the gap, call for a reconsideration of the order parameter of Sr$_2$RuO$_4$.
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