Physicists dating back to Feynman have lamented the challenges of applying the variational principle to quantum field theories, most notably evaluating and optimizing expectation values of a quantum field state. In the context of non-relativistic quantum field theories, this approach requires one to parameterize and optimize over the infinitely many n-particle wave functions comprising the state's Fock space representation, a seemingly daunting task. In this work, we introduce a variational ansatz to enable the application of the variational principle to 1D bosonic quantum field theories directly in the continuum. Our ansatz is a neural-network quantum state, and uses the Fock space representation to model a quantum field state as a superposition of n-particle wave functions, each of which is parameterized by a common neural-network architecture that is both permutation-invariant and able to accept an arbitrary number of arguments. We develop a novel algorithm for variational Monte Carlo in Fock space and employ it on our ansatz to approximate ground states of the Lieb-Liniger model, the Calogero-Sutherland model, and a regularized Klein-Gordon model. Our ansatz can be seen as the neural-network-based analog of continuous matrix product states, which have traditionally been deployed on 1D field theories but struggle on inhomogenous systems and long-range interactions. The utility of our ansatz lies in its flexibility and broad applicability to such systems, providing a powerful new tool for probing quantum field theories.