Theoretically, yes, a state can exist in the absence of a system of states. However, in practice, it’s possible to question whether a state would have to exist minus a system of states.
One could argue that a state doesn’t need other states to be a state. By definition, a state is a structured political community with a distinct government. Such an entity doesn’t require preexisting states to function. Any given populace or territory can act as a state regardless of the surrounding system. They can pass laws, care for their citizens, dispense justice, and so on even if no other states are around.
In the concrete world, it might be hard to claim that a state can exist absent a system of states. A state achieves power and recognition from other states. A state exists to interact with these other states. Minus a system of states, the state—as a tangible political body—has no technical need to exist. In other words, the state lawfully exists because of the system.
In the United States of America, states exist to be part of the system of states and the power that such a system confers. Right now, Washington DC is not a part of the system of states. The district behaves like a state—it has a local government, police force, schools, and so on—but it’s not recognized as a state in the system. It doesn’t have representatives who can vote in the House or Senate.
If Washington DC wasn’t a part of a system of states, there’d be no clear need for it to officially become a state. Since Washington DC is a part of a system of states, some leaders believe that it needs to be granted statehood. With Washington DC, one can see why a state might not need to formally exist absent a system of states and, conversely, why a state might need to officially exist in relation to a system of states.