The meaning of terms like "apposition/appositive" seem to be debated, so it might not be the clearest way to describe this construction.
To me, the construction that you are asking about seems to be along the lines of things that are analyzed as reduced relative clauses or the product of "whiz-deletion": "A man the size of a giant" and "Somebody her age" can be rephrased as "a man who was the size of a giant" and "somebody who was her age".
The definition of "apposition"
"Apposition" might be taken to refer to replaceability
One user of this site, BillJ, uses a definition of "apposition" that seems to require that the appositive NP could replace the first NP. That doesn't work in your sentences, which don't entail that "The size of a giant came up to me" and "Her age shouldn't do such exercises". You can see more discussion of this definition of apposition in the comment thread here: What is the grammatical term of "whose wife was a school teacher"?
Or apposition might be considered to be associated with non-restrictive constructions specifically
"Appositional constructions", a 2011 thesis by Herman Heringa, mentions a distinction between "close" apposition, which does not involve parenthetical commas, and "loose" or "non-restrictive" apposition: Heringa says that the latter is "usually taken as apposition proper" (p. 3). Consequently, Heringa only analyzes the non-restrictive construction in his thesis, although he does mention that Meyer (1992) has a different viewpoint, seeing restrictive and non-restrictive constructions as different types of a single phenomenon (p. 3).
Heringa suggests looking at De Vries (2008a) for "more details on the restrictive construction" (p. 5).