Everything has a mass, but it's very difficult measure the lower mass.

A string is a particle at a lower level than the concept of mass. Current theory says that mass derives from the Higgs field: massive particles interact with this field (by exchanging Higgs Bosons), whereas massless particles do not. Strings would constitute all particles, massless or not. If the vibrations of the string are such that it interacts with the Higgs field, it shows the behaviour that we call mass; if not, then not. Your original statement does not make sense. It would imply that every massive particle would be capable of infinite subdivisibility. Whatever you divided the first chunk into would also have mass, therefore (by your account) volume, and thus could be subdivided again. However, I do not thing this is regarded as true. Strings occupy space, but cannot be divided. Also, there is a limit (the Plank Length) on how small things can be: you cannot divide things below this size.

A string's mass depends on its vibrational mode, and only certain modes are possible. It is also not quite right to say that if something has mass, it must have volume. An electron is treated, in quantum mechanics, as though it has no volume inside it, yet it has a mass.

"A string is simply a one dimensional extended object. However, it is not like just an ordinary string, say a violin string. The energy density along a string in the fundamental string theory is assumed to be a universal constant which is usually denoted as 1/2pa ¢c, where c is the velocity of light and a ¢ is the new fundamental constant characterizing string theory. Thus, even when the string stretches or shrinks, the energy per unit length does not change. In other words, the total mass of a string is essentially determined by its total length. This means that the length of the string in the states with lowest energy is zero, at least classically, and so the masses of these states vanish. If we treat the string quantum-mechanically, we have to take into account quantum fluctuations and therefore we cannot say that the length of the lowest energy states of the string is strictly zero in the classical sense. However, the fact that their masses are vanishing is still valid. It also turns out that the states have in general nonzero ‘spin’, namely they are rotating. The massless states of closed strings, strings which close upon themselves, necessarily contain a spin-2 state, using some appropriate unit for measuring the strength of rotation (called angular momentum). The massless states of open strings, strings with open ends, similarly contain a spin-1 state. The spin-2 massless close string state behaves as graviton, which is responsible for the universal gravitational force. In the low-energy limit, it exactly coincides with the graviton one expects from the quantization of general relativity. The spin-1 open string state coincides with the gauge particles, like photon corresponding to the electromagnetic interaction. Basically, this is why string theory in general contains gravity and/or gauge forces." "the time–energy uncertainty relation can be reinterpreted as the uncertainty relation of the space–time" "It clearly indicates that the strings cannot probe short-distance scales to arbitrary precision with respect to both time-like and space-like distances, simultaneously. It is appropriate to call the relation as the space–time uncertainty relation. To appreciate the significance of this simple-looking relation, we should remember that, due to the selfcontainedness of string theory, the space–time structure itself should be determined self-consistently by using the dynamics of strings themselves. The above relation must then be interpreted as the qualitative characterization of space–time itself, if one believes string theory as the fundamental unified theory. We may express this situation by claiming that space–time is ‘quantized’." Source and further information: http://www.iisc.ernet.in/currsci/dec252001/1554.pdf