To boldly go where everyone has gone before...

Let's be real, galactic colonization is not going to happen anytime soon. 

1. There's no hyperdrive/warp. The physics we have today say it's impossible. Say, in a few years / decades we figure out that it is possible, we'd have nuked / diseased ourselves into oblivion, never mind the Grey Goo apocalypse. 
2. If there were, even going at X times c where X is a triple-digit number would still take years, maybe decades of real time, shipboard, not relativistic time to get there. Which means the ship is effectively a generation starship, which means the mass of all things needed to sustain the humans aboard outweighs the humans aboard by 25 gajillian tons to the kilo. Plus the colonization stuff. Even if the drive did ALL the moving point A to B, just think of the logistics of getting it into the ship and down again. 
1. Cryosleep? Still not as cost-efficient. 
3. Ok, so don't send humans. Send a digitized bank of DNA codes for everything needed, a nanotech factory, and an AI / semi-AI to make it all happen. Will fit under a couple of dozen kilos, but still leaves the rest of us right here. 
4. No hyperdrive, no space elevators, no all-powerful benevolent AI to stop us from nuking / diseasing each other into the void, sea levels rising... we have a few decades at best, and time's running out. 
5. Terraform the solar system? Absolutely. It's hedging bets, and we have the tech (or will have in five years) the means to get there. Maybe in 50 years, even the means to survive there, in a place with different gravity, different soil, different air, different temperature... assuming any of them exist. Stretch goal, but doable. 
6. Space stations, then? As the Earth fades away after the asteroid impact / Singularity event / zombie apocalypse / nuclear winter, there will be a dozen to a few hundred little lights floating in orbit or L1 to 5 points, maybe even the moon... recycling air, water, food... racing on their little centrifuges to make sure their children gestate normal and their bones don't turn to jelly... 
7. Leaving the final, most economical, most doable answer: 

Vaults. 

Weight is not a problem. Space is not a problem. Go deep enough, and soil is not a problem. Gravity is not a problem. With efficient scrubbers, even air, water, and minerals are not a problem. All you need is a solid door and a long, long time. 

Dozens, hundreds, thousands of vaults. 

With functional ecologies, seed or DNA banks, cryosleeping residents tended by robots, or even - as the ultimate backup - that same nanotech factory and DNA bank buried deep underground or in near-Earth orbit with a thousand-year alarm clock ticking away... 

We're here. Whatever happens, we will survive. The planet may die, but we will remain.