We study how a Yukawa coupling of the Standard Model fermion to a light scalar effects the stellar structure of cold stellar remnants such as neutron stars.  We use a simple model of a massive scalar coupled to a single dominant fermion species with no other interactions. For a broad scalar mass range, the equation-of-state and stellar structure depends only the effective coupling $g_{eff}=g_f m_f / m_\phi$, where $g_f$ is the Yukawa coupling, $m_f$ the fermion mass and $m_\phi$ the scalar mass. If the effective coupling g_{eff} is larger than O(1) the Yukawa coupled matter exhibits various anomalous behaviors including hydrodynamic instability, negative pressure, distinct phases (soft and hard) of matter with sharp phase boundaries between them and with vacuum.  These anomalies can lead to stars exhibiting an unusual mass-radius relation compared to that predicted by Oppenheimer-Volkoff model. To the extent that these anomalies have not and/or will not be observed limits the size of the new Yukawa coupling.