Hayflick limit and hair or nails
Let's do some math.

Also, let's ignore your examples, and use a clearer one - instead of fingernails, let's look at the cells that line the intestine.  Intestinal epithelial cells are shed and replaced constantly and very rapidly, roughly every 5 days.  So that means about 50-70 cell generations per year.  

Each time the cell replicates, it loses a little bit off the end of its telomeres.  How much does it lose?  

We know that for lymphocytes, adults lose roughly 30 base pairs of telomeres per year.  Lymphocytes don't replicate as fast as the colonic mucosa, though.  The estimate for these rapidly-dividing cells is about 44 bp/year. We can say 50 for a round number.  

That doesn't tell us much unless we know what we're starting and stopping with.  Humans start life with about 10,000 bp of telomeres.  At least in cultured cells, once telomeres reach 1000-2000 bp, cells become senescent and stop, or greatly reduce, their growth rate. 

So we start at around 10,000, we can lose up to 9000, and we lose around 50 per year. That means that the intestinal epithelium, which have about the most rapidly-dividing cells in the body, are good for about 180 years of life, assuming that the loss remains constant.

(I don't know if loss does remain constant in these cells, but in lymphocytes, the loss of telomeres accelerates in elderly people, from about 30 to about 60 bp/year in people over 60. However, it's likely that reflects increased lymphocyte turnover in older people, not something intrinsic to the age per se.)

So bottom line, we're good for it.  

More generally, keep in mind that the popular media have seized on telomeres as a simple answer for everything related to aging.  Obviously, that's not reality.  Telomeres certainly reflect aging; in some cases, perhaps many cases, they may be one influence on aging; but in most cases, telomere shortening is probably only a minor aging factor, and many other causes are more important.