I would like to understand in which cases the word "distance" is a mass noun and in which it is not. Indeed, distance is something that one can measure, so for me it looks like it should be a mass noun. However, one says,
Please keep your dog at a distance.
Question 1. What is the rule here?
Question 2. Are there some similar situations in which a "naturally mass noun" becomes countable?
The word distance is meaningful in both an uncountable sense and a countable sense. So, the type of determiner (or lack of any determiner) that you precede it with, or whether you use the plural, indicates how you want the listener to understand it.
First, here is uncountable distance:
Let's put some distance between us and Houston. [That is, let's drive away from Houston.]
The force of gravity is inversely proportional to the square of distance.
It wouldn't make sense to put "distances" in either of those sentences.
The countable sense of distance is any specific distance, like 2.1 meters, 20 miles, or 3 inches. You can count these:
This star map shows the distances from Earth to 2,000 different stars. [The distance from Earth to Alpha Centauri, the distance from Earth to Sirius, the distance from Earth to Polaris, etc.]
My first day on the job as a surveyor, I measured 47 distances.
No one would really say the second one, but it does make sense.
Once you understand countable distance, then you can understand common expressions like at a distance and a distance.
I watched the battlefield from a safe distance. [The distance between me and the battlefield was great enough that the bullets couldn't reach me.]
The arc of a baseball in flight isn't caused by a force that Earth exerts on the ball at a distance, as Newton's theory held. Instead, the ball is responding to the gravitational field immediately around it. [Source: Spooky Action At a Distance (2015). Action at a distance is direct causal influence by one object on another object that is spatially separated from it. In other words, there is a non-zero distance between the objects.]
Please keep your dog at a distance. [Please keep your dog far enough away from me that I don't need to fear getting bitten.]
These countable uses of distance don't specify a distance numerically. They refer inexactly to a distance: whatever is far enough for the purpose under discussion, even if only implicitly, as in the sentence about the dog. However, it would be strange to put "distance" into the plural in any of these last three sentences.
The same applies to other uncountable nouns whose meaning can reasonably be seen in a countable way. You can put an indefinite article in front of them or put them in the plural to lead the reader to understand them in a countable sense, the exact meaning determined by context and custom:
When a woman is past a certain age, you no longer ask her her age.
Did an alien intelligence probe planet Earth in 1954?
The strengths of the wrestlers were so unequal, there was never any doubt who would win the match.
Once upon a time, there was a President Pro Tempore who presided over the Senate for a time, that is, until the Vice President returned. [The first "a time" denotes a certain moment in (fictional) past time; the second "a time" denotes a certain time interval.]
During the low-water season of each year, the rock reef, acting as a natural barrier, caused a substantial part of the water of Fall River to flow upstream into Pitville Pool. During floods, the waters of the two rivers flowed over the reef, leaving very little of Fall River impounded in the pool. [Source: Water Rights Laws in the Nineteen Western States (1974). Plural "waters" often refers to turbulent water on a large scale, as in a flood or where rivers meet.]
Social and economic pressures often drive professionals to become blind to elements of their field of expertise that are obvious to outsiders, e.g. economists who don't see that people choose jobs for reasons other than salary or linguists who deny that sentences communicate thoughts.
There is no general rule about what is correct, beyond what you can reasonably expect a reader to understand and what phrases involving the noun have meanings well established by previous usage.
Very interesting question. I don't think the inclusion of the indefinite article here is not so much making distance a countable noun – although I can understand how a learner might parse that sentence and draw that conclusion.
Instead, the phrase at a distance is a standard expression – one that gets its own entry in some dictionaries.
For example, from Macmillan:
at/from a distance (phrase)
at/from a place that is not close
I’ve only ever seen him from a distance.
Tim followed him at a distance (=keeping a long way behind him).
and from Collins:
at/from a distance (phrase)
If you are at a distance from something, or if you see it or remember it from a distance, you are a long way away from it in space or time.
The only way I can cope with my mother is at a distance.
But I think your question is even more interesting because distance (like many other nouns) can be used in both a countable and uncountable sense. If you look through the definitions in Macmillan, you'll see the UNCOUNTABLE label used six times, but COUNTABLE label used twice.
This is a case where you can't really judge the noun's countability based on the articles around it.
When a mass noun denotes a quality of something (as opposed to a substance), the same word is often used to denote a specific occurrence, appearance, or measurement of that quality, or multiple of those in the plural.
For example:
"You need height to play basketball" vs. "Their heights would make them a good basketball team".
"The humiliation was unbearable" vs. "I was subjected to 6 distinct humiliations that evening".
"Absence makes the heart grow fonder" vs. "Sometimes an absence is as notable as a presence".
"energy is force times distance" vs. "A transferal of energy may result from a force applied through a distance".
"Mercy must be freely given" vs. "Her many small mercies sustained me".
When someone says "keep your dog at a distance", they are using the word in that second sense, referring to the specific distance between themselves and your dog. It is a short form, though, in which they neglect to characterize the distance. Of course there is always a distance between the person and the dog, but the person wants that distance to be a safe distance, a significant distance, a respectful distance, a distance of at least 5 meters, or maybe just a distance that doesn't make them afraid.
The person doesn't want to bother with the significant introspection that would be required to precisely specify the kind of distance they need, so it's just a distance. Just keep the dog away, OK?
