We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an $\widetildeΩ(n)$ lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [Frishknecht et al., SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show $\widetildeΩ(n)$ lower bounds for computing $(\frac{3}{2}-\varepsilon)$-approximations of the diameter or the radius, and for computing a $(\frac{5}{3}-\varepsilon)$-approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [Frishknecht et al., SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of $(α,β)$-spanners, for $α< β+1$.