In 2008, a class of binary sequences of period $N=4(2^k-1)(2^k+1)$ with optimal autocorrelation magnitude has been presented by Yu and Gong based on an $m$-sequence, the perfect sequence $(0,1,1,1)$ of period $4$ and interleaving technique. In this paper, we study the 2-adic complexities of these sequences. Our results show that they are larger than $N-2\lceil\mathrm{log}_2N\rceil+4 $ (which is far larger than $N/2$) and could attain the maximum value $N$ if suitable parameters are chosen, i.e., the 2-adic complexity of this class of interleaved sequences is large enough to resist the Rational Approximation Algorithm.