Sure the set has no supremum or infimum in the rational number system, but if that was meant to be answered within the rationals, it really shouldn't have been defined as the set of rationals between "sqrt2 and sqrt3", but rather the set of "positive rationals x with x^2 between 2 and 3", since several logical and philosophical objections arise automatically from the former verbatim.

Considering this I personally think the set was meant to be viewed as a subset of real numbers, and not in the rational system. In this case the infimum and supremum sure dn exist and finiding them is not much hard.

However, I suggest you should refer to the teacher/ professor/ textbook that had proposed the question for an explicit refrence to the system on which the exterma are to be found.