How far away is the nearest star after the Sun?

Proxima Centauri, about 4.24 light years - roughly 40 trillion kilometres. Voyager 1, the fastest object humans have sent out of the Solar System, would take something like 70,000 years to cover that distance if it were pointed there, which it is not.

It is a red dwarf, far too faint to see without a telescope, and it belongs to the Alpha Centauri system - a triple star, with two sun-like stars orbiting each other and Proxima loosely bound to them at a considerable distance. Whether Proxima is genuinely part of the system was uncertain for decades and was only settled with precise velocity measurements.

It has planets, including Proxima b, which is roughly Earth-sized and in the zone where liquid water could exist. The complication is that red dwarfs are violent: Proxima produces enormous flares, one recorded in 2019 brightening it by a factor of 14,000 in a few seconds, and a planet close enough to be warm is close enough to have its atmosphere repeatedly stripped. It is the nearest place worth asking the question about, and the answer is probably no.