Can the gravitational force be described just with virtual particles of its own field? Similarly to steady electric or magnetic fields for example?
 A: Lets clear up terminology:
Virtual particles are mathematical entities in Feynman diagrams, used in quantum field theory in order to calculate measurable quantities, crossections, lifetimes etc.

The wavy line represents the virtual particle , in this case the photon,and it is a mathematical propagator where the mass of the virtual particle is in the denominator and the line carries the quantum numbers of the named particle, except that its four vector is off mass shell, it is not unique, it varies  under the integral limits.
A steady electric or magnetic field is a classical measure. One can connect it with QFT since classically an electric field is measured by a test charge, so that one can write a feynman diagram between the particles of the source of the field ( in the above case the electron on the left) and the test particle ( the electron on the right). There is an isomorphism which when cumbersomly carried through the QFT mathematics gives the same answer for the value of the electric field.  So in a sense the electric field can be associated with the mathematical virtual particles.
Now these electrons have a gravitational field. The expectation of physicists is that an equavalent diagram can be written for the gravitational exchange of a graviton, BUT there is no definitive quantum field theory of gravitation. There do exist effective field theories, which presume that the infinities from higher order diagrams will be someday resolved, and who do use Feynman diagrams for gravitational interactions.
In this framework  the classical gravitational field will emerge from the quantum mechanical mathematical exchanges of virtual gravitons, similar to the way electric fields emerge from the quantum mechanical mathematical exchanges of virtual photons. It remains to be seen whether expectations will be fulfilled. String theories have no infinity problems with gravity, so maybe in the future  the hypothesis will be justified, if they lead to the theory of everything.
A: Virtual particles are used as a tool in Feynmann diagrams for describing phenomena which happens between real "physical" objects (e.g. particles). So you can not describe a theory in physics without having as ingredients the physical objects = real particles.
