In mountainous Greece (the islands and mainland are both very mountainous) there are often springs with pure drinking water where local people come to fill up bottles and jugs to take home and even mule herds or what have you, bring animals to drink. These are usually by the side of the road and the more water an area has the more of these springs or pee-ghee there are. Streets in Athens and all over Greece are named after springs usually called just Zodogou Pigi or life giving spring.
In some larger towns, spring water is sometimes piped in from some distance and to little taps with ornate, often arched housings around them sort of like the one photo right.
Springs like these are not for tourists only but preceed the concept of tourist as we know it. These springs are for the local people and their animals so don't be afraid to partake of a centuries old tradition of sharing and distributing water.
Loose in the enviroment, in their own milieu, wiser tourists are often seen with bottled water, sometimes pulling one out of their day pack, and rightly so, as museum and archeological site water is questionable, rarely free and needlessly expensive. Is it an accident or did they plan it that way? My guess is better than yours and I'm leanin towards: its no accident!
By Law bottled water comes in 2 sizes LG and SM. and costs 40 or 50 eu cents for a small and 1eu for a one liter size or the large.
Depending on where you are in MAINLAND Greece, taverna and restaurant tap water is perfectly safe to imbibe and often preferable. There is likely no need to order bottled water. Local water is perfectly safe and drinkable in many towns especially in mountainous areas. Essentially all of Greece is mountainous.