The philosophers stone also known as "the great work" was the search not so much for eternal life, or to turn lead into gold, but the search for perfection, for God.
Turning a human into an immortal represented a step closer to our original pre-fall condition, and turning lead into gold meant transmuting a base metal closer to perfection, gold being seen as the most perfect, the most god-like of all the metals. (Alchemists did in fact accidentally invent gold plating in the process.)
This search to turn lead into gold has, in modern times, become the symbol of the foolishness of the alchemists and of the middle ages in general... except they were absolutely right, metals can be transmuted and base metals can be turned into gold. Of course to do so, the alchemists would have needed a nuclear reactor.
The transmutation of metals is not uncommon. It happens in nuclear reactions both natural such as at the center of stars (the idea that lead can be turned to gold in the center of a star would have appealed greatly to the alchemists) and in man-made nuclear reactions such as in particle accelerators like the LHC and in conventional fission power reactors.
In 1980 at the Lawrence Berkeley Laboratory, Glenn T. Seaborg, a nobel prize winning chemist, achieved for the first confirmed time in history man-made transmutation of a base metal, bismuth, into gold. Granted it was only several thousand atoms, not even enough to see with the naked eye, and cost many many times more to create then it was worth, but nonetheless 'the great work' had been achieved. As it turns out, the philosophers stone is a well funded national laboratory.
(There is a rumor of a Soviet nuclear research facility, sometimes identified as the "Soviet nuclear center of Lake Baikal," where in 1972, an experimental reactor lead shielding was found to have been transmuted to gold. Unfortunately there is little in the way of documentation to support this story.)
There is a further irony, this time on the alchemists. It is in fact much easier to turn gold into lead, then the other way around and gold left in a nuclear reactor will eventually turn into lead.
But then, Alchemists promising to find a way to turn gold into lead just doesn't have the same ring to it.