What does it mean to have a QFT that can not be encoded by an action. What is by far the most powerful approach of study in such a case. What is the best studied physical theory that falls into this category. What is QFT? I think it is a set of rules that facilitate descriptions of things we can observe, but what is the most mathematically accurate way of capturing what it is? I have read of cases where the path integral approach fails what does this mean? Is there any geometric structure whose properties describe the space of all posible QFT's? What happens when you can't use an action?
It is possible to abstract the notion of a QFT away from the notion of Lagrangians/Hamiltonians, one axiomatic way are the Wightman axioms. As one can see, they reduce the quantum theory to its very heart: A Hilbert space where the states live and a field operator that acts upon it, generating "particles", all of this happening in a Lorentz covariant way.
Concrete example of QFTs without an action are CFTs in 2 dimensions, they are almost completely fixed by demanding that they are a QFT in the axiomatic sense which has a Virasoro symmetry.
The "space of all possible QFTs" is quite a complicated (and open, I think) question, since it is quite difficult to prove that a given QFT is really a QFT in the sense of the Wightman (or Osterwalder-Schrader) axioms.
A large class of QFTs called 2 dimensional conformal field theories, can be defined without an action.
The general method for constructing an any dimensional CFT is as follows, you can calculate the action of the symmetry generators of the fields using Ward identities and their commutation relations given by the Virasoro algebra which allows you to construct the whole hilbert space from the knowledge of how your set of "primary" fields transform under conformal transformations which is encoded in the conformal dimension. Why you can do this in CFTs but not in any QFTs is the state-operator correspondence. Each state of the Hilbert space is in 1-1 correspondence with the set of field operators. There is a subset of these called minimal models which are defined through a finite set of primary fields. One can even give them a Lagrangian formulation through the Couloumb gas formalism which consists of coupling it with the Ricci scalar.
What distinguished a d-dimensional CFT to a CFT in 2 dimensions is that the conformal group of a 2D CFT is infinite dimensional. Any locally holomorphic map corresponds to a local conformal transformation. This allows you to use Laurent expansions and other tools of complex analysis.
For "What is a QFT"? This page from nLab maybe useful. It somehow "axiomatizes the assignment of algebras of observables to patches of parameter space".
For further info. on CFT refer to di Francesco et Al's Conformal field theory.