You can get four solutions. However, the solutions will be effectively useless, and the conditions under which they apply will be unreadable.
%solving fourth order algebraic equation to get g
syms g y z
x = 0.0585;
n = 0;
Pi = sym(pi);
eqn = 1/g-sqrt(1 + z.^2/((2*n+1)*Pi*y + 4.4*Pi*x*g).^2) == 0;
sol = solve(eqn, g, 'returnconditions', true, 'maxdegree', 4);
G = simplify(sol.g)
C = simplify(sol.conditions)
Furthermore...
solve() is for finding indefinitely precise solutions. However, your input value 0.0585 is not indefinitely precise, instead representing some value between 5845/100000 (inclusive) and 5855/100000 (exclusive). It does not make logical sense to ask for exact solutions when some of the inputs are known precisely known. There are y, z values for which this makes a difference. Quartics can be very sensitive to exact values in determining which parts are real valued or which parts are complex valued.
