A magnetometer using spin-polarized K and 3He atoms occupying the same volume is used to search for anomalous nuclear spin-dependent forces generated by a separate 3He spin source. We measure changes in the 3He spin precession frequency with a resolution of 18 pHz and constrain anomalous spin forces between neutrons to be less than 2x10(-8) of their magnetic or less than 2x10(-3) of their gravitational interactions on a length scale of 50 cm. We present new limits on neutron coupling to light pseudoscalar and vector particles, including torsion, and constraints on recently proposed models involving unparticles and spontaneous breaking of Lorentz symmetry.