Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distance $d$ and dual distance $d^{\perp}$. In recent years, much research has been devoted to improving the lower bound on $d$, some of which have exceeded the square-root bound. The constructions presented recently seem to indicate that when the minimum distance increases, the minimum distance of its dual code decreases. In this paper, we focus on the new constructions of binary cyclic codes with length $n=2^m-1$, dimension near $n/2$ and both relatively large minimum distance and dual distance. When $m$ is even, we construct a family of binary cyclic codes with parameters $[2^m-1,2^{m-1}\pm1,d]$, where $d\ge 2^{m/2}-1$ and $d^\perp\ge2^{m/2}$. Both the minimum distance and the dual distance are significantly better than the previous results. When $m$ is the product of two distinct primes, we construct some cyclic codes with dimensions $k=(n+1)/2$ and $d>\frac{n}{\log_2n},$ where the lower bound on the minimum distance is much larger than the square-root bound. When $m$ is odd, we present two families of binary $[2^m-1,2^{m-1},d]$ cyclic codes with $d\ge2^{(m+1)/2}-1$, $d^\perp\ge2^{(m+1)/2}$ and $d\ge2^{(m+3)/2}-15$, $d^\perp\ge2^{(m-1)/2}$ respectively, which leads that $d\cdot d^\perp$ can reach $2n$ asymptotically. To the best of our knowledge, for the binary cyclic codes with length $n=2^m-1$ and dimension $k=(n\pm1)/2$, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound.