The graphs below show the pull of two springs versus their stretch. In each case their spring constants are 2 Newtons per cm. The upper plot shows a "zerolength" spring. When totally relaxed, this spring would have zero length. Obviously this is impossible, because at some point the coils will touch each other and the spring will not be able to shorten any more. For example:

This spring can't contract any more because the coils are in contact with each other. This is a necessary condition for a zerolength spring, but not sufficient. The spring is zero length only if a plots of force versus length extrapolates back to zero force at zero length. 
Let's say that the spring above is 5 cm long when relaxed and that it is a zerolength spring with a spring constant of 2 Newtons per centimeter. If the spring is stretched an additional cm so that it is 6 cm long, the force will be 6 times 2 equals 12 Newtons.
A "normal" spring with a spring constant of 2 Newtons per centimeter is shown in the lower graph. When relaxed it's coils are not touching each other and its length is 5 cm.
If extended to 6 cm length, the pull would be (6 minus 5) times 2 equals 2 Newtons.
Let's see what's special about a "zerolength" spring. The diagram above shows that for the selected mass (M) the boom is horizontal at equilibrium. Then the torque from the spring must equal the torque from the pull of gravity (Mg). Lets compute what the mass must be for this condition.
MgA = kS(Y/S)A = KYA
M = kY/g
Lets check the torques in this position:
Due to the mass: torque = MgA Cos(theta)
Due to the spring: torque = kS' Y Cos(theta)A/S' = kYA Cos(theta)
What mass will be in equilibrium in this position?
MgA Cos(theta) = kYA Cos(theta)
M = kY/g
These calculations show that the same mass (M) will be in equilibrium at any position. This is equivalent to saying that the restoring force is zero, or the period is infinite. Obviously this isn't exactly what we want for seismology, but with a slight adjustment of the spring, the the period can be made very long.
Comments on actually making and using a zerolength spring by Allan Coleman.