From what I understand of the question, there are three interlinked concepts that are causing confusion, each of which could be a separate question.
1. Is axial stress dependent on length of a specimen?
Axial stress is independent of length. @Wasabi's answer goes into more detail, but the core concept is that a stress of $\sigma$ on a specimen of length $L$ has the same effect on a specimen per unit length as a stress of $\sigma$ on a specimen of length $2L$, or indeed of any length.
2. Is axial strain dependent on length of a specimen?
Axial strain is independent of length. This follows from the answer to (1.) above, and from Hooke's law, specifically that
$$
\sigma = E \varepsilon
$$
so that if $\sigma$ is independent of length, so must be $\varepsilon$ since they are linearly related.
3. Is axial deformation dependent on length of a specimen?
Axial deformation is dependent on length. It also depends on cross-sectional area (assuming constant cross-section) and tensile modulus. This follows from a quick and dirty derivation of the axial deformation formula. Consider a constant-cross-section tensile member of area $A$, length $L$, experiencing a uniform load $P$, with modulus $E$. The member will experience a deformation of $\delta$. Note that by definition $\varepsilon = \delta / L$ and $\sigma = P / A$.
From Hooke's law,
$$
\sigma = E \varepsilon
$$
and substituting our notes gives
$$
\frac{P}{A} = E \frac{\delta}{L} \\
$$
which after some rearranging yields
$$
\delta = \frac{PL}{AE}
$$
which is the equation for axial deformation of constant-cross-section tensile members. As should be obvious from the equation, axial deformation depends on length, as well as area.
How Can I Think About This Intuitively?
Elasticity is just like springs, and indeed Hooke's law is also used for linear springs. Now consider two springs such that one has twice the number of coils as the other, but the rest of their parameters are the same. Assume they are also weightless. If you held just the end of each spring, hooked identical weights to the other ends and let them uncoil toward the floor, the spring with twice the number of coils would extend twice as far. As a result, the change in length per per unit original length, equivalent to the strain, would be the same.
The same is true of two tension rods, one twice as long, experiencing the same stress with the same cross-sectional area. The one that is twice as long would have double the deformation, but they would have the same strain.
How Can I Use This Information?
As an example, consider a tensile rod. Suppose you must minimize axial deformation. Then you would choose a material with high tensile modulus so that stress has less effect on strain, and thus less effect on deformation.
Now suppose instead you have a family of tensile rods all made of the same material. If each must not deform more than 1 mm, then the longer rods must have a larger cross-sectional area to make up for the additional "slack" from their added length.
What About Plastic Deformation?
The above three questions only apply in the elastic regime. In the plastic regime, volume is practically conserved in most materials, which means that any instantaneous extension of a tensile specimen must cause an instantaneous reduction of area such that the volume gained by increasing length is equal to the volume lost by reducing the cross-section area.
However, there is an instability inherent to tensile plastic deformation called necking, which you noted. Namely, if one length-wise segment of a specimen is slightly narrower than the rest of the specimen, stresses will be concentrated at the narrow region and it will reach the elastic limit slightly before the rest of the specimen. Plastic deformation will thus begin at the slightly narrower section, causing a greater decrease in area than the rest of the specimen, further concentrating stresses, which causes further and greater deformation. The specimen will tend to fail wherever it is narrowest, and will do so unstably in a smooth, constant-cross-section specimen. That is, it is impossible to predict where failure will occur without intentionally notching the specimen.
Beware! Necking does not necessarily proceed the same way in polymers. In polymers, necking is associated with polymer chain alignment, which locally strengthens the material so much it prevents further deformation there, unlike metals. Instead, the neighboring regions begin to neck causing them to strengthen, and so on, until the entire specimen becomes aligned.