There are no real solutions. First, I converted them to a problem that fimplicit can use.
F1 = @(x,y) -(118/121)*x.^2-(333/242)*y.^2-(196/121)*x.*y+(63/121)*x-(361/242)*y-(724/121);
F2 = @(x,y) (21/121)*x.^2+(299/484)*y.^2+(80/121)*x.*y-(35/242)*x+(391/484)*y-256/121;
So the same equations, changed using find and replace in the editor to insure no mistakes were made.
H2 = fimplicit(F2,[-500,500,-500,500]);
Your expression F(2) has a locus of solutions that form a hyperbola. Two hyperbolic arcs, extending to infinity in either direction.
The first however, has no real solutions at all. We can prove that using solve. For example, converting F1 to a symbolic form, if I use solve to find the two roots of the quadratic for X as a function of Y, we will find it involves a square root, of an expression that looks like this:
sqrt(-2.7438*Y^2 - 7.5058*Y - 23.069)
The sub-expression inside the sqrt is negative for ALL values of Y. Therefore all solutions for F1 are complex. It has no real solutions that will make it zero, for any inputs.
fplot(-2.7438*Y^2 - 7.5058*Y - 23.069)
As I said, the solution involves the squre root of that value, which is always negative, as a parabolic arc.
As far as having received an error, the message was a correct assessment, although that was actually a warning message, not a true error. There simply are no real solutions to this set of equations.