The results of a numerical simulation of electron field emission from the surface of 2-D fractal object bounded by a Julia set are presented. The calculations show that emission occurs mainly from small parts of the surface close to selfsimilar small-scale irregularities, where the electrostatic field is enhanced. The results presented confirm the hypothesis, that the unusual field-emission properties of some materials can be explained by taking into account the fractal structure of its surface. The connection between the fractal dimension and the emission properties are discussed. It is found that the emission current depends mainly on the geometry of small-scale irregularities. Nevertheless, the fractal dimension defines the slope of current-voltage characteristics.