What nonsense. (Admittedly sarcasm is not a much better response than supercilious dismissal.) Infinity as a mathematical object, not merely some 19th century epsilon-delta definition, crops up all over in mathematics and there are several senses in which the "square root of infinity" makes sense. Just because the real number line doesn't include "infinity" as a number doesn't mean the question has no answer.
The surreal numbers include a host of transfinite numbers, and not just square roots but arbitrary powers are defined on them. (These can be identified as infinite two-player games as well...)
However, it's easy enough to define a multiplicative semigroup (or monoid) that includes infinity, and note that "inf * inf = inf" so inf is its own square root, which is no more contradictory than '1' being its own multiplicative identity. You just have to admit that it's no longer a group.
IEEE 754 floating point arithmetic has a similar property; try squaring numbers with big exponents and you'll quickly get "infinity" as an answer. These "square roots of infinity" are pretty numerous.