It's no trick at all to obtain a Poisson ratio of nearly zero in a material in general; you just need to suppress any mechanisms that would produce a lateral expansion (or, much less typically, contraction) upon uniaxial compression.
If we remove material, for example, then we sidestep the issue—that we'd otherwise encounter—that uniform condensed matter tends to stay at a constant volume when it's deformed (particularly so for liquids and compliant solids). Cellular solids, however, i.e., solids with some missing regions (such as foams, honeycombs, cancellous bone, and cork) can easily deform uniaxially with negligible lateral deformation. Consider the uniaxial crushing of an aluminum honeycomb as an extreme example in which a cellular solid easily accommodates densification:
Papka, Scott D., and Stelios Kyriakides. "In-plane compressive response and crushing of honeycomb." Journal of the Mechanics and Physics of Solids 42.10 (1994): 1499-1532.
An excellent general reference on this topic is Gibson and Ashby's Cellular Materials; in fact, the authors begin by describing Robert Hooke's ca. 1660 observations of the microscopic structure of cork, a type of oak bark:
Micrographia (1665) https://www.nlm.nih.gov/exhibition/hooke/images/cork1.jpg
(Such observations ultimately led to the term "cell" for the proliferating biological unit of life, as the spaces reminded Hooke of a monk's monastery cell.)
Micrographs of cork from a modern scanning electron microscope strongly resemble Hooke's sketches:
Fortes, M. A., and M. Teresa Nogueira. "The Poisson effect in cork." Materials Science and Engineering: A 122.2 (1989): 227-232. Left image shows a section parallel to the radial direction, revealing staggered cell closures separated by corrugated walls. Right image looks down the radial direction, showing approximately hexagonal divisions.
It's useful to refer to the conventional coordinate system:
Gibson, Lorna J., and Michael F. Ashby. Cellular Solids: Structure and Properties. Cambridge university press, 1999.
The cork cells are closed, roughly hexagonal cells (as viewed down the radial direction) with corrugated walls (as viewed in the axial or tangential directions) that collapse easily under load. As a result, Poisson's ratio involving the radial direction is very small and nearly zero because the solid material simply folds up into the existing adjacent spaces. (Other related outcomes are that Young's modulus and the bulk modulus are low, the density is low, the thermal conductivity is low, and the loss coefficient is high, as is the energy absorption capacity.)
Greaves et al.'s review article on Poisson's ratio $\nu$ plots this parameter against the ratio of the bulk modulus $B$ to the shear modulus $G$:
Greaves et al. "Poisson's ratio and modern materials." Nature Materials 10.11 (2011): 823.
As expected (and as discussed in Gibson and Ashby's textbook), we find that other cellular materials such as honeycombs and auxetic materials with reentrant structures share the $\nu\approx 0$ space adjacent to cork.
Four years after the OP, this question still doesn't have an answer, yet it comes up in the first page of google searches on fabric stretch, so I will give it a shot.
Why should adding a small amount of a stretchy material make an
otherwise non-stretchy fabric stretch? Shouldn't the non-stretch
fibres still constrain the maximum stretch of the fabric?
You are completely correct in your speculation: adding stretchy material to an otherwise non-stretchy fabric does not increase the stretch of the fabric at all. As you suggested, the amount of stretch is completely determined by the length and alignment of the the least stretchy fibers in the weave or the knit.
Instead, stretchiness is increased by making the least-stretchy fibers longer and distributing the extra length in some geometry that allows them to be pulled straighter when tension is applied in the direction of interest. The ability to distribute extra fiber using many different geometries is one of the primary contributions of knitted structures compared to woven ones.
There are two reasons why high-stretch fabrics contain added elastic material. The most important purpose is to improve the recovery of the fabric when the tension is released. The same knitted or woven fabric without the elastomer would stretch just as much, but would be baggy and would recover very slowly, or may not recover until processed in some way (washing, steaming, ironing). Adding elastomer in the right amounts and the right places allows that same fabric to spring back immediately.
The second way in which elastomers improve stretchable fabrics is by adding resistance to stretch. Where a stretchable fabric without elastomers may be stretched with very little force, adding elastomers increases the amount of tension required to stretch the fabric. This allows the fabric to retain its shape and structure until a designed-in amount of tension is applied.
Best Answer
Idealized liquids have a Poisson's ratio of exactly 1/2 and are not incompressible (because no material is incompressible). No solid can have a Poisson's ratio of exactly 1/2.
To explain: If you apply a deviatoric (i.e., not hydrostatic; the 1-D version is shear) load on a liquid, it will deform without resistance and without a change in volume, implying that $\nu=1/2$. One way to see this is to use one of the elasticity relations (which assume a homogeneous isotropic linear elastic material): $G=\frac{3K(1-2\nu)}{2(1+\nu)}$. All stable materials have a positive bulk modulus $K$ (i.e., all stable materials compress to some degree under hydrostatic stress); thus, setting $\nu=1/2$ implies a shear modulus $G$ of zero, which corresponds to a fluid.
Let's look at a couple of the elasticity relations from the Poisson's ratio side: $\nu=\frac{3K-E}{6K}$, where $E$ is the Young's elastic modulus, and $\nu=\frac{3K-2G}{2(3K+G)}$. Thus, elastomers have a Poisson's ratio of nearly 1/2 because their shear and Young's elastic moduli are much smaller than their bulk moduli. When you shear or pull on rubber, for example, it's relatively easy to unkink and uncoil its long polymer chains to obtain shearing or uniaxial deformation. When you apply pressure from all sides, however, you're essentially trying to push C atoms closer to C atoms, which is not easy.
Note that this review (Greaves et al., "Poisson’s ratio and modern materials", Nat Materials 10 2011, DOI: 10.1038/NMAT3134) defines $\nu=\frac{3K-2G}{2(3K+G)}=0$ for gases for unexplained reasons, possibly because they are idealizing gases as exhibiting $K=0$.