Ask yourself if you want your Fock space to itself be a Hilbert space.
Now look at the so called "finite-particle space" which is ... by definition ... a subset of the set of all functions from $\mathbb N$ into the union of all the finite products of 1d particle spaces. Specifically it is the subset where each function sends each n into the space of products of n single particle states and such that the function gives zero for all but finitely many $n\in \mathbb N.$ That's just what the words mean (and you cab restrict to where the finite products are symmetric or antisymmetric as needed).
So we clearly know what this "finite-particle space" is and it is clearly a vector space, and we can give it an inner product, but it is most definitely not a Hilbert space. Can we think of this vector space as a subspace of a larger space that is a Hilbert Space?
Yes. The Fock space. You can think of the Fock space concretely as the topological/metric closure of the "finite-particle space" as an inner product space.
Fine. But why was this so confusing? And what was going on with the notation? The notation and terminology can be understood from the perspective of category theory. If you take the algebraic category of vector spaces and take the direct sum you got the "finite-particle space" and if instead you took the Hilbert Spaces in the topological category of Hilbert Spaces and take the sum you get the Fock space.
In both cases you can think of them as the smallest algebraic type vector space capable of holding any of the spaces (the algebraic sum) versus the smallest topological type vector space (i.e. Hilbert Space) that is capable of holding any of the multiparticle spaces.
So if you think of them as vector spaces and take the sum you get the "finite-particle space" and if you think of then as Hilbert spaces and take the sum you get the Fock space.
It just comes down to whether you want the sum to be a Hilbert space or just a vector space. If you want it to be a Hilbert space it has to be a bit bigger to fill in the holes the "finite-particle space" leaves