Continuous gravitational waves are nearly monochromatic signals emitted by asymmetries in rotating neutron stars. These signals have not yet been detected. Deep all-sky searches for continuous gravitational waves from isolated neutron stars require significant computational expense. Deep searches for neutron stars in binary systems are even more expensive, but these targets are potentially more promising emitters, especially in the hundreds of Hertz region, where ground-based gravitational-wave detectors are most sensitive. We present here an all-sky search for continuous signals with frequency between 300 and 500Hz, from neutron stars in binary systems with orbital periods between 15 and 60 days and projected semimajor axes between 10 and 40lt-s. This is the only binary search on Advanced LIGO data that probes this frequency range. Compared to previous results, our search is over an order of magnitude more sensitive. We do not detect any signals, but our results exclude plausible and unexplored neutron star configurations, for example, neutron stars with relative deformations greater than 3x10^-6^ within 1kpc from Earth and r-mode emission at the level of {alpha}~a few 10^-4^ within the same distance.