In phrase "The distance from Moscow to Kazan is 500 miles" which word is the most appropriate for translating "distance" - "расстояние", "дистанция", "дальность", or "даль"?
The best is "расстояние", in the majority of cases this is the most appropriate one.
"Дистанция" is more of something related to measurements in sports and in military related terminology, for instance:
- Спортсмен преодолел пол-дистанции очень бодро, но всё-таки уступил лидерство.
- С обеих сторон выпустили массу ракет с большой дистанции с нулевым результатом.
"Дальность" is mainly used with fixed small set of nouns, the main usage is "дальность стрельбы", which is "shooting range" in English - and that actually gives pretty clean idea what is "дальность" - it's range.
"Даль" - it looks like you are relying on Google Translate, which indeed translates it as a distance. Well, it is, but only in phrases like "всмотреться в даль" - "to peer into the distance". So, distance here is not a measurement of any kind, it is a concept of something that is remotely located.
"Расстояние", most definitely.
"дальность" is range, and the term is most usually applied to the moving objects which can cover the range, less often to the static objects in range. In a latter case it will rather have specific military use - "цель на дальности x" - target at x distance.
"даль" is a more abstract and poetic term, "farness". Not used to represent a numerical value of the distance.
"дистанция" is mostly used in sports, maybe also some technical context.