I hold a faculty position in statistics (that's for the AC above who called me a "passer-by sitting at home in their boxers munching on Hot Pockets", so I guess I have to pull credentials, though in his defense my post sounded more dismissive than what I'd wanted).
Yes, the p-value threshold of 0.05 is considered "standard" in many applied sciences, in particular medicine. It is convenient for many of reasons that were outlined by other posters (cost, number or persons required for an experiment, ethics). It does not mean that it is intellectually satisfactory. The joke among statisticians is that this value was introduced about 100 years ago by the R.A. Fisher (one of the founding fathers of statistics) who once wrote something akin to "if we decide on a value of alpha such that the probability of falsely claiming a discovery when the null hypothesis holds seems reasonably low, say for instance, alpha=5%...", and this has somehow been engraved as gospel ever since.
The truth is, this threshold value of 5% is now considered very lax by modern statisticians, essentially because of the very large numbers of published papers reporting significant values as compared to Fisher's times. The posts of penguinoid and ras above explained it very professionally, one can also refer to "Why Most Published Research Findings Are False" (Note: this was published in PLOS medicine, hardly an obscure journal)
In conclusion, my post was certainly not a defense of soda pop (there is already sufficient evidence that it is extremely damaging for your health for very clearly identified reasons), but a reminder that the specific results of this study (the effect on telomeres), though certainly not to be dismissed, should not be considered as established truth at this point, but rather pointing in a direction which should be investigated further for confirmation. That, by the way, is the actual meaning of "being skeptical", unfortunately this tends to be conflated with "being in obtuse denial" nowadays.