You have a ‘stiff’ system, and ode45 is not the best option for it, although it's an appropriate initial experiment. If ode45 seems not to work (and the system otherwise seems to be solvable in some situations as yours was), switching to ode15s is appropriate, especially in systems with coefficients of widely-ranging magniudes such as yours are (from 4E+1 to 1E+9).
After experiencing the same problem you did with ode45, I ran your code with ode15s and it solved your eoms system in 1.2225 seconds.
So I suggest you simply change your ODE function call to:
[t, x] = ode15s(@eoms, tspan, x0);
and enjoy the results!
Also, I coded your function as an implicit function to run it, avoiding the need to save a separate function m-file. (This isn't always possible, but it works in many situations.)
eoms = @(t,x) [1e9 + 5.0e4*x(3) - 50*x(1); 4.0e1*x(1) - 3.3e3*x(2); 2.0e3*x(2) - 5e4*x(3) + 3.5e7*heaviside(t-1)*x(4); 1.0e3*x(2) - heaviside(t-1)*5.0e7*x(4)];
and the call to it changes to:
[t, x] = ode15s(eoms, tspan, x0);