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Various quasi-zero stiffness (QZS) systems have been developed and applied in the vibration control domain in recent years. However, most QZS systems are usually unstable against external disturbances, and the QZS ranges of them are very limited. To address the above issues, this study develops a highly stable QZS vibration isolation system integrated with magnetorheological fluids (MRFs). The MRFs endow the vibration isolation system with stiffness variability in vertical and lateral directions against external disturbances, which innovatively solves the unstable problem of the QZS system. Meanwhile, the stiffness variability also makes the system adaptable to vibrations with different frequencies, so the system can deliver the best vibration isolation performance in response to various excitations. The system consists of a vertical isolation unit and a lateral isolation unit. Through paralleling a nonlinear positive stiffness QZS component with a nonlinear negative stiffness QZS component in the vertical isolation unit, a large QZS range in the vertical direction and smaller stiffness are realised, thus improving the vibration isolation performance. In this study, the vibration isolation system is designed and prototyped; its QZS characteristics and adjustable stiffness features in both the vertical and lateral directions are experimentally verified; the frequency responses of the system are obtained experimentally; the stability and the vibration isolation performance of the system are also evaluated by experiments. This study provides a solution to overcome the unstable problem of QZS systems and extend the limited QZS range, whilst realising QZS characteristics in both vertical and lateral directions, thus broadening the application of QZS systems.