It is actually pretty simple to solve this. Remember that fsolve uses a basic algorithm. Start the problem out in the wrong place though, and it will fail. I compare it to the idea of starting a blind person out at some random place on the surface of the earth, and then asking them to find some spot, using only a cane. (Disclaimer: No blind people were harmed in this thought experiment!) If you start the person out in a difficult to escape place, they will not succeed in your task.
I'll do part of this problem symbolically, because it will make things simpler to show you.
syms x y
u = [300; 1/60];
F(1) = (618830617597725.*y)./18014398509481984 - (21108551597470609.*x)./90071992547409920 + u(1)/5;
F(2) = ((2382265234661919.*x)./4722366482869645213696 - (2382265234661919.*y)./4722366482869645213696 + u(2).*((- y.^4 + (6707.*y.^3)./100 - 2121.*y.^2 + 8532.*y)./(200.*(y.^4 - (671.*y.^3)./10 + 2396.*y.^2 - 31350.*y + 936000)) - (6375194751874021.*y)./4722366482869645213696 + 37./10000))./((4.*y.^3 - (20121.*y.^2)./100 + 4242.*y - 8532)./(800.*(y.^4 - (671.*y.^3)./10 + 2396.*y.^2 - 31350.*y + 936000)) + ((4.*y.^3 - (2013.*y.^2)./10 + 4792.*y - 31350).*(- y.^4 + (6707.*y.^3)./100 - 2121.*y.^2 + 8532.*y))./(800.*(y.^4 - (671.*y.^3)./10 + 2396.*y.^2 - 31350.*y + 936000).^2) + 6375194751874021./18889465931478580854784);
First, solve the first equation for x, as a function of y. This one is just linear, so easy to solve.
x1 = solve(F(1),x)
(1031384362662875*y)/7036183865823536 + 112589990684262400/439761491613971
F2 = subs(F(2),x,x1);
substitute that into the second equation, and plot. I'm doing this for a reason, so that you will understand how a solver gets hung up, NOT how you should necessarily have solved the problem. It would have been a great deal of help regardless.
The curve seems to go to -inf as y grows large. with one root at
y0 = fzero(matlabFunction(F2),300)
But there is also a flat, shallow valley on the left, but it never crosses zero there.
The problem is though, IF you start the iterations out below around y at 150? Then the solver will get hung up in that valley to the left. This is as if you had started the blind sercher out in the vicinity of the Dead Sea, and asked him to find the lowest spot on Earth. Is it even conceivable they will manage to climb uphill to get out of the local depression you put them in, and then manage to wander to the other side of the Earth to find the Marianas trench?
Now, where did you start out fsolve?
Yep. You started things out in a bad place. fsolve never had a chance to achieve success.
Could you make things simple enough to solve? Well, yes. Reduce the problem to one variable as I did. One variable problems are MUCH easier to solve than are two variable problems. Now, with a little effort, you will be able to make fzero find the root consistently.