In: Physics
Can general relativity be completely described as a field in a flat space? Can it be done already now or requires advances in quantum gravity?
ADDITION: in the paper by Straumann, N. - Reflections on Gravity (ESA-CERN Workshop on Fundamental Physics in Space and Related Topics, European Space Agency, 5-7 April 2000(2001), SP-469,55), it is shown (similarly to Deser's classical work but in a more expository style) that a spin-2 field theory on a flat Minkowki 4-manifold ends up being totally equivalent to Einstein's curved spacetime where the ab inition Minkowski metric ends up being unobservable whereas the observable and physical metric is dynamical supporting the relational meaning advocated in the rest of this answer which emphasizes the idea that space, time and causal structure are relational notions between dynamical entities and NOT an absolute static stage (be it Newton's or Minwkoski's) where things, including gravity, live.
Conclusion from the long digression below: yes, you can formulate gravity as a field in a flat manifold (e.g. Deser's, Doran-Gull-Lasenby, tele-parallelism...) but no, that manifold (and its additional structure) is not flat space-time. In any equivalent formulation of general relativity, you cannot avoid the coupling between the physical space-time metric and causal structure to the rest of the degrees of freedom (matter, forces...). Therefore, in any formulation, there are fields living on top of each other, and space-time amounts to relational coincidences between them. The underlying manifold serves as a necessary indexing device for the degrees of freedom of fields, but has no physical observable meaning. If the situation is such that the region of interest can be coordinatized by physical observables (e.g. matter fields embodying observers in that region), then we may use these as indexing the fields' coincidences; if moreover the coupling of those observables to gravity can be neglected, then they describe a physical Minkowskian space-time in that region. (But Minkowski's flat space-time is not a solution to Einstein's field equations when there is a non-vanishing cosmological constant! so assuming physical reality to a global flat 'space-time' is meaningless in any equivalent formulation).
For a wonderful discussion and development of all these conceptual issues, you should read Carlo Rovelli's - Quantum Gravity, chapter 2 (in special 2.2, 2.3 and 2.4).
Einstein's general relativity is a theory about the dynamics of space-time; more precisely, it is the realization that the gravitational field 'is' space-time, and that non-relativistic (newtonian and minkowskian) space and time is a particular solution useful as background in regimes of approximately neglectable gravitational effects compared to the other phenomena of interests. As Einstein remarked in his original article,
"... the requirement of general covariance takes away from space and time the last remnant of physical objectivity..."
A. Einstein, Grundlage der allgemeinen Relativit