For answering the question, it makes sense to look at the definition of the term polymer as recommended by the International Union of Pure and Applied Chemistry (IUPAC) that they give in their Gold Book. It says a polymer is...
A substance composed of macromolecules.
So the word polymer does not refer to individual molecules although it is colloquially used like that. Then, a macromolecule is defined as:
A molecule of high relative molecular mass, the structure of which essentially comprises the multiple repetition of units derived, actually or conceptually, from molecules of low relative molecular mass.
The concept of the repeating units is already described in the other answers. Here, I would like to focus more on the question what is meant by a high molecular mass. This is covered by the first note on the IUPAC macromolecule definition:
In many cases, especially for synthetic polymers, a molecule can be regarded as having a high relative molecular mass
if the addition or removal of one or a few of the units has a negligible effect on the molecular properties. [...]
So focussing now on your example, we can have a look at the boiling point of some alkanes:
||boiling point [°C]
As you can see, the physical property of the boiling point changes considerably with the number of carbons per molecule and with the molecular mass, so these low molecular mass alkanes are no macromolecules, and an ensemble of these molecules will not be a polymer.
Now, for the example of polyethylene I found a nice paper, unfortunately behind a paywall. It shows a graph of melting temperatures of n-paraffins and polyethylene as a function of number of carbons per molecule. The melting temperature inceases at low carbon counts, but levels off around approx. 1,000. This would correspond to a polyethylene macromolecule with 500 repeating units with a molecular mass of around 14,000 g/mol which could be considered the lower molecular mass limit for polyethylene. This limit is far from being a strict limit, there is always some ambiguity, but for all practical purposes this is sufficient.
When I was a student, we were taught that as a rule of thumb, macomolecules will have molecular masses above around 20,000 g/mol, so the estimation above fits in quite well.
Molecules with lower molecular masses which still are composed of repeating units would be called oligomer molecules, and ensembles of these molecules would be called oligomer. So an alkane like dodecane could be considered as a low molecular mass oligoethylene.
 J. Phys. Chem. 1965, 69, 2, 417–428