The "infinite tensor product" picture may be useful as a sort of concrete image of the state space of a quantum field theory, but in practice is rarely used because of the technical difficulties it brings along. Moreover, the paragraph from Streater and Wightman you quoted could be read as saying that infinite tensor products of Hilbert spaces could be thought of as a bigger mathematical "receptacle" for the actual "natural state space" of the theory, but even that may not be actually tenable.
Let me unpack my last sentence above. First of all, what do Streater and Wightman mean by the "natural state space" of the theory? This must be read in the context of the subject of their book as a whole. In physical terms, they actually mean "states accessible from the vacuum state through local operations", or at least to a finite but arbitrary degree of accuracy. Implicit in this sentence is the assumption that the vacuum state is unique, which may not always be the case. Let us assume for the time being that it is, though (we will return to this point later). Mathematically speaking, this sentence summarizes the content of the Wightman(-GNS) reconstruction theorem, which tells us how to recover this Hilbert space from the vacuum expectation values of field products smeared with Schwartz test functions in space-time, seen as distributions (as mentioned by jjcale in his comment to the OP). This Hilbert space can be seen as the completion of the linear span of all smeared field products applied to the vacuum state vector.
In the case of free fields, this relates to the canonical field quantization as a continuum of oscillators corresponding to the different Fourier modes once you write the field operator smeared with a Schwartz test function in space-time in terms of these oscillators. The Hilbert space you obtain in this fashion is the same. Streater and Wightman's sentence "some of its observables involve all the oscillators at once" can be understood as a consequence of the uncertainty principle: if you smear your field product with test functions supported in a bounded region of space-time, representing the smearing in momentum space using the Fourier transform shows that all field oscillators contribute to the observable since the Fourier transform of these test functions vanish in no nonvoid open subset of 4-momentum space if they are not identically zero.
All seems well and good, but does the reconstruction theorem recover all physical states? The answer is unfortunately no, not even within finite but arbitrary accuracy. A typical example of a state not accessible from the vacuum state by means of local operations within arbitrary precision is a thermal equilibrium state of finite, nonzero temperature with respect to some Lorentz frame. The same can be said about any state reached from the latter state by means of local operations - they live in "disjoint" Hilbert spaces. Moreover, there are theories where the vacuum state is not unique, and with distinct vacua not locally accessible from each other in the above fashion. It is by no means clear whether all these states fit into an infinite tensor product of Hilbert spaces such as the the one in the OP or not.
Of course, you could try to perform instead a "big" direct sum / integral of all these disjoint Hilbert spaces and use that as your state space receptacle, but this space (if it can be constructed at all in this fashion) is bound to be also technically extremely unwieldy to use. If you need to work with disjoint states in the above sense, is technically and conceptually better to work instead with the algebraic concept of state: as a certain kind of linear functional on the algebra of local observables of the theory and perform the GNS construction with respect to a convenient reference state (e.g. the vacuum or more generally a thermal equilibrium state with respect to some Lorentz frame) when a concrete Hilbert space is needed. All the information you need for that is already in the algebra of local observables. In other words, you no longer treat all states of the theory as belonging to a single Hilbert space but only look at subspaces of "mutually locally accessible" states, so to speak. These subspaces are usually called sectors of the theory. Under fairly general assumptions such as the ones in the Wightman-Garding formalism, sectors are always separable.
More broadly, the basic tenet of the so-called algebraic approach to quantum field theory (AQFT) by Haag and Kastler is that all the physical information of a quantum field theory is contained in how the algebras of local observables associated to each space-time region (generated in our present context by field products smeared with test functions supported in that region) embed into each other. This is the appropriate language to deal with families of disjoint physical states, in whatever physical context they may appear (thermal equilibria, superselection sectors, etc.).