We propose that a strongly interacting particle is a finite region of space to which fields are confined. The confinement is accomplished in a Lorentz-invariant way by endowing the finite region with a constant energy per unit volume, B. We call this finite region a "bag." The contained fields may be either fermions or bosons and may have any spin; they may or may not be coupled to one another. Equations of motion and boundary conditions are obtained from a variational principle. The confining region has no dynamical freedom but constrains the fields inside: There are no excitations of the coordinates determining the confining region. The model possesses many desirable features of hadron dynamics: (i) a parton interpretation and presumably Bjorken scaling; the confined fields are free or weakly interacting except close to the boundary; (ii) infinitely rising Regge trajectories as a consequence of the bag's finite extent; (iii) the Hagedorn degeneracy or limiting temperature; (iv) all physical hadrons are singlets under hadronic gauge symmetries. For example, in a theory of fractionally charged, "colored" quarks interacting with colored, massless gauge vector gluons, if both quark and gluon fields are confined to the bag, only color-singlet solutions exist. In addition to establishing these general properties, we present complete classical and quantum solutions for free scalars and also for free fermions inside a bag of one space and one time dimension. Both systems have linear mass-squared spectra. We demonstrate Poincaré invariance at the classical level in any dimension and at the quantum level for the above-mentioned explicit solutions in two dimensions. We discuss the behavior of specific solutions in one and three space dimensions. We also discuss in detail the problem of fermion boundary conditions, which follow only indirectly from the variational principle.