There may not be a direct way, however, you can derive the forward kinematics to give you more insight in how the links are operating with respect to one another. I have worked out the forward kinematics below:
For this particular serial linkage robot, I locked joints 3 and 4 to stay at 0 degrees since this will maximize reach as far as possible (obtained by observation).
On the bottom right are the X, Y and Z equations. Just stick a point in 3D space that your wanting to reach in the X, Y, and Z variables.
Lets call (L2 + L3 + L4) = a
This will give you:
X = a*c1*c2
Y = a*s1*c2
Z = a*s2
Lets say I wanted to know what the cumulative link length should be if I wanted to reach point (300, 300, 300):
300 = a*c1*c2
300 = a*s1*c2
300 = a*s2
Because there are 3 equations and 3 unknowns, we could try systems of equations, after which I obtain:
Theta1 = 45
Theta2 = 35.2644
a = 519.615
Which means you can play around with link lengths L2, L3, and L4, but they must combine to equal 519.615.
Note: Because we are dealing with something as nonlinear as sines and cosines, there may be more solutions to the equation than what I provided.
Overall, messing around with the end equation I provided by selecting various points in 3D space you want to reach or selecting angles of interest for theta 1 and theta 2 is the best way I can think of to determine link lengths.
Another thought could be to use the equations I provided to create an ellipsoid/sphere that touches your farthest points you can reach based on your link lengths; don't know quite how I would go about doing that of the top of my head though.