If you know that ε is a rational number, then you do not need inapproximability for Max-2-SAT to prove your statement. A typical proof of the NP-hardness of Max-2-SAT (e.g., the one in the textbook Computational Complexity by Papadimitriou) actually proves the NP-completeness of L1/5. To prove the NP-hardness of Lε for positive rational numbers ε<1/5, we can reduce L1/5 to Lε as follows: given a 2CNF formula φ (an instance for L1/5), let m be the number of clauses in it. Let r and s be positive integers such that (1/5−ε)mr = 2εs holds. Then construct a 2CNF formula (an instance for Lε) by repeating φ for r times and adding s pairs of contradicting clauses. A simple calculation shows that this is indeed a reduction from L1/5 to Lε.
This reduction clearly works only if ε is rational, because otherwise r and s cannot be taken as integers. The general case where ε is not necessarily rational seems to require inapproximability, as Yuval Filmus wrote in his answer.