Reconfigurable intelligent surfaces (RISs) have drawn much attention recently for their appealing advantages in shaping wireless channels to improve the spectral and energy efficiencies of wireless communications. However, conventional fully-passive RISs generally suffer from the so-called ``multiplicative fading'' effect which thereby limits RISs' practicability and manufacturability. In this paper, a novel architecture of ``Reconfigurable Distributed Antennas and Reflecting Surfaces (RDARS)'' is first proposed to overcome this limitation from the ``multiplicative fading'' effect. Specifically, unlike existing active RIS variants, RDARS inherits the low-cost and low-energy-consumption benefits of fully-passive RISs by default configuring all the elements as passive to perform the reflection mode. On the other hand, based on the design of the additional direct-through state, any element of the RDARS can be dynamically programmed to connect with the base station (BS) via fibers and perform the connected mode as remote distributed antennas of the BS to receive signals. Consequently, a controllable trade-off between the reflection gain and the distribution gain can be achieved via RDARS at the BS. To unveil the system behavior of the RDARS-aided system, we analyze the received signal-to-noise ratio (SNR) under maximum ratio combining (MRC) at BS. Closed-form outage probability and ergodic achievable rate are also provided and are verified through extensive simulations. To demonstrate the superiority of the proposed RDARS, experiments are carried out using a prototype of RDARS with a total number of 256 elements which revealed extra 76% throughput improvement could be achieved by deploying RDARS with only three elements performing connected mode. This thus confirms the effectiveness of the proposed RDARS and envisions it as a promising candidate for future 6G wireless systems.