The issue is that with the choice of units, an order-1 guess for c[1] causes cosh and sinh to overflow. (At first I thought these must be CGS units, but in that case, g = 9.81e02 cm/s^2, right?)
julia> m, g, L0 = 257*2.7, 9.81e03, 600e03;
julia> x1, x2, y1, y2 = 0, 500e03, 0, 0;
julia> c = [1;1;1];
julia> m*g/c[1]*(x1 + c[2])
6.807159000000001e6
julia> cosh(m*g/c[1]*(x1 + c[2]))
Inf
You can prevent the initial overflow by setting the guess for c1 to something large, e.g.
julia> c = [1e06;1;1];
julia> cosh(m*g/c[1]*(x1 + c[2]))
452.14957457816104
But a nonlinear solver is going to have a hard time dealing with a search space where there are large differences of scale on the components of c. Better to nondimensionalize the system or choose units where your numbers are order 1 or order 10. MKS should do.