How long does a linear hydrocarbon chain have to be before we call it polyethylene?
Actually, polyethylene is still an alkane as it fits IUPAC definition perfectly fine [1, p. 1313]:


  alkanes:
  Acyclic branched or unbranched hydrocarbons having the general formula $\ce{C_nH_{2n+2}}$, and therefore consisting entirely of hydrogen atoms and saturated carbon atoms.


Smaller alkane molecules you are referring to, are oligomers. IUPAC defines the prefix oligo- as follows [1, p. 1353]:


  oligo:
  A prefix meaning 'a few', and used for compounds with a number of repeating units intermediate between those in monomers and those in high polymers. The limits are not precisely defined, and in practice vary with the type of structure being considered, but are generally from 3 to 10.


At the same time removing numerical restraints when referring to oligomer molecules [2, p. 2289]:


  1.2 oligomer molecule
  A molecule of intermediate relative molecular mass, the structure of which essentially comprises a small plurality of units derived, actually or conceptually, from molecules of lower relative molecular mass. 


Obviously, the boundary between oligomers and polymers is blurred. Usually for the emergence of "polymer" sufficient degree of polymerization $N$ starts with $10^2 - 10^4$ for synthetic polymers and about $100 - 200$ for globular proteins.

A significant feature of any polymer as compared to its molecular weight counterpart is that when one or several units are added or removed into the polymer molecule, the properties of the polymer material remain almost unchanged. Properties of oligomers substantially depend on molecular weight.

As a typical example, melting points of low molecular weight alkanes gradually increases with polymerization degree (Source):


  


reaching a plateau of approx. $\pu{400 K}$ for linear low density polyethylene.

References


Moss, G. P.; Smith, P. A. S.; Tavernier, D. Glossary of Class Names of Organic Compounds and Reactivity Intermediates Based on Structure (IUPAC Recommendations 1995). Pure and Applied Chemistry 1995, 67 (8–9), 1307–1375. https://doi.org/10.1351/pac199567081307.  
Jenkins, A. D.; Kratochvíl, P.; Stepto, R. F. T.; Suter, U. W. Glossary of Basic Terms in Polymer Science (IUPAC Recommendations 1996). Pure and Applied Chemistry 1996, 68 (12), 2287–2311. https://doi.org/10.1351/pac199668122287.