Cavity-enhanced radiation-pressure coupling between optical and mechanical degrees of freedom allows quantum-limited position measurements and gives rise to dynamical backaction, enabling amplification and cooling of mechanical motion. Here, we demonstrate purely dispersive coupling of high-Q nanomechanical oscillators to an ultrahigh-finesse optical microresonator via its evanescent field, extending cavity optomechanics to nanomechanical oscillators. Dynamical backaction mediated by the optical dipole force is observed, leading to laser-like coherent nanomechanical oscillations solely due to radiation pressure. Moreover, sub-fm Hz(-1/2) displacement sensitivity is achieved, with a measurement imprecision equal to the standard quantum limit (SQL), which coincides with the nanomechanical oscillator's zero-point fluctuations. The achievement of an imprecision at the SQL and radiation-pressure dynamical backaction for nanomechanical oscillators may have implications not only for detecting quantum phenomena in mechanical systems, but also for a variety of other precision experiments. Owing to the flexibility of the near-field coupling platform, it can be readily extended to a diverse set of nanomechanical oscillators. In addition, the approach provides a route to experiments where radiation-pressure quantum backaction dominates at room temperature, enabling ponderomotive squeezing or quantum non-demolition measurements.