Of the three linguistic entities in this sentence, only one is well defined. Let's rephrase from a question to a statement and look at the veracity: "Real numbers are real".

- "Real Numbers" - this is the one thing we can formally define, but there is implied baggage - does the concept of real numbers exist, or does at least one real number exist? The former is just ∃x(x ∈ ℝ), √2 proves that, the set ℝ must exist, no major problem there. Really this question asks "For ∃x(x ∈ ℝ), is x 'real, existing in reality'?" For that we need to look to the other two words.

- "are" - the copula (is/to be) is a doozy and a perennial semantic footgun, put a pin in that

- "real" - this is a nebulous concept, and where the weight of the discussion lies. Basically it is asking can something with infinite information be? And there's that pesky copula.

Is/be is tricky, particularly in English, because its usage is heavily overloaded. It is used to express

- Strict equality: "Two plus two is four"

- Categorization: "A frog is an amphibian" (all frogs are amphibians, not all amphibians are frogs)

- Description: Grass is green (grass has the property green, but green is not grass).

- Predicates (in general): "It is raining" (in many languages, there is no copula, and the sentence is closer to "Raining." or "It rains." see Zero Copula [0]) (arguably, all the above are involved in predicates, definitions are hard)

In "Real numbers are real", I would contend we are talking about the description usage or predicate. Real numbers have the property of "existence". This is where a messy problem gets ultra messy. But at the same time, I believe the problem more broadly collapses to "Does X Exist?" [1] for any X. Chairs are made up of components which do not individually have anything chair-like about them, but the arrangement of matter (the process of producing the chair) nonetheless has the description of chair-like-ness. Real numbers comprise a process which produce them (stating a mathematical formula and asking what the answer is), and thus the process defines their existence. Whether or not an object with infinite information exists is irrelevant if the process to produce that object is finite, and that object can be used to produce another answer (e.g. we can use √2 to define the dimensions of the A Series of paper, or broadly solve for the dimension of a 45° triangle with side length 1). Its existence can no more be denied than the existence of chairs, or heaps, or other intangible ideas.

(I suppose this argument only applies to constructable/algebraic numbers, which means the original question is really only talking about transcendental reals with no "nice" expression. But that gets into Hilbert's seventh problem, and whether you can derive an expression of any arbitrary transcendental using two algebraics).

"Reality" as a linguistic concept is also highly relative. Are dreams real? They are experienced, they can be measured with medical equipment, and they can have lasting psychological side-effects; they are "real". But the events of the dream do not themselves have the observed side-effects in "consensus reality"; they are not "real".

tl;dr - language is nebulous, and nebulosity is challenging [2]

[0] - https://en.wikipedia.org/wiki/Zero_copula

[1] - Do chairs exist? - Vsauce - https://www.youtube.com/watch?v=fXW-QjBsruE

[2] - https://meaningness.com/nebulosity