Parametric modulation in nonlinear dynamical systems can give rise to attractors on which the dynamics is aperiodic and nonchaotic, namely, with largest Lyapunov exponent being nonpositive. We describe a procedure for creating such attractors by using random modulation or pseudorandom binary sequences with arbitrarily long recurrence times. As a consequence the attractors are geometrically fractal and the motion is aperiodic on experimentally accessible time scales. A practical realization of such attractors is demonstrated in an experiment using electronic circuits.