Just to be clear: In general, you do not use an article with a proper name in English.
Where you do put "a" before a proper name is when you want to indicate that all you know about the person is his name. If someone you know who is named Bob Smith is selected for the job, then you say, "Bob Smith has been selected." If you read the name in some sort of announcement and have no idea who this person is, then you might say, "Someone named Bob Smith has been selected", or "A Bob Smith has been selected." Often in that case you put the name in quotes. This would also apply if there are several people with the same name and you are not sure which it is. "Who was selected?" "A Bob Smith. But I'm not sure if it's the Bob Smith from the English Department or the Bob Smith from the Math Department."
You almost never use "the" in front of a person's name. If you do it's usually when you are specifying just which of several people with similar names you mean. Like in my example above about the two Bob Smiths. Or if you want to make clear that you mean the famous person by this name. Like, "We got THE Bill Gates to speak at our conference", meaning, the actual famous person of that name, not just someone else with a similar name.
Like most words, space has more than one meaning, and we must use contextual reasoning to work out which is intended.
You're thinking of the meaning of space which is a mass noun: that is, uncountable, which is why the article seems misplaced to you.
For example, this is how Merriam-Webster (MW) defines the uncountable space you're thinking of:
space (n, uncountable): the boundless three-dimensional extent in which objects and events occur and have relative position and direction
However, it is the very presence of the article in your text which serves as a clue that that's not the definition being employed here.
To the contrary, because there is an article, the intended meaning must be some kind of count noun. Thus, we refer back to our dictionary, and look for definitions of space which not only fill that role in the sentence (i.e. be count nouns), but also make sense in context.
To remind us of the context:
A person who suffers from claustrophobia has a dread of being confined in a small space,
So the space under consideration must be limited (or the claustrophobe could not be confined to it), and it must be possible to have more than one of them (again, the article establishes that we're dealing with a count noun).
So, returning to Merriam-Webster:
space (n, countable): an area that is used or available for a specific purpose
That looks quite suitable: first, it is a count noun, because it's described as an area (did you notice that area can be a count noun too?), and second, it is set aside for some particular purpose, which in our case, is to confine the claustrophobe.
We can get further corroboration by looking at the subordinate glosses:
a limited extent in one, two, or three dimensions : distance, area, volume
an extent set apart or available
Again, we see articles before extent in both subdefinitions, and now we're explicitly told that the space is both set apart and, crucially, limited. That is, it's capable of being small.
The key takeaway here is not that space can take an article sometimes, but the more important lesson that when you're surprised or confused by a particular usage, it's always best to first check a dictionary to see if there are any senses for the word you were previously unaware of.
And I can tell you that this happens even to native speakers, and we use the same strategy. Surprised (baffled!) that anymore can be used in a positive context? Check a dictionary and learn that one of its meanings is nowadays.
Best Answer
One could say:
The students in Mr Smith's art class were at their easels, looking at the sunset.
In that case, the sunset is the object of scrutiny, and it refers to the entire western sky as it is lit up by the setting sun.
But when it is used to indicate a time of day, the idiom is "before sunset", "after sunset", "at sunset".
EDIT: CopperKettle's examples on the use of "the" with a proper noun are good ones. Here are some others in the same vein; perhaps we can extract the essence from them to show when it is appropriate to use "the" with the proper noun.
I do not know you any longer! What has happened to the gentle Henry Jekyll, the Henry Jekyll with the wry sense of humor and a fondness for good port?
Get up off your ass, Jones, and get back on that horse! Where's the do-or-die Jones, the gung-ho Jones, the let-me-at-them Jones who volunteered for this mission??? I don't like this new sissy Jones who is afraid to ride on a pony just because it's a little skittish.