10^-35 meters

More at the Planck Length

The following table provides the fundamental Planck quantities with their approximate SI values.

Quantity	SI value
Planck length	1.62 x 10^-35 m
Planck time	5.39 x 10^-44 s
Planck mass	2.18 x 10^-8 kg
Planck temperature	1.42 x 10³² K

Gravity is a classical theory in that it allows point sources, infinitely divisible to a true dimensionless point. Gravitational theory works well with large-scale structures but the theory provides meaningless answers at ultramicroscopic scales. On the other hand, the uncertainty principle of quantum mechanics requires fluctuations in the structure of spacetime near the Planck scale.

To probe smaller and smaller scales, higher and higher energies (smaller wavelengths of photons) are required. At the Planck scale, a limit is reached requiring concentrations of energy so great that they collapse into black holes.

I recommend the following sources for more information on the incompatibility of General Relativity and Quantum Mechanics at the Planck scale:

Just Six Numbers, Martin Rees
Constants, John D. Barrow
The Fabric of the Universe, Brian Greene
The Elegant Universe, Brian Greene
Cycles of Time, Roger Penrose
Three Roads to Quantum Gravity, Lee Smolin