No, the stars are not out of reach. You just need to change your planet-centric bias and have a VERY large ship. You may also need fusion power, but fission might work, depending on the composition of free bodies. (I.e., asteroids, planets, etc. not bound to any star.) You also need a very long trip time, as you don't want there to be an unreasonable delta-V when you send out mining parties while on the way.
The problem, if such it is, with this approach is that by the time the party reaches the next star system, they'll see no reason to bother with the planets, and just build another ship or so to allow population growth...and then head off for parts unknown.
OTOH, you will need a large population, and robots, etc. The machines will need to be maintained, redesigned, rebuilt, etc. And this means a full range of highly technical skills.
The problem with this approach is that there is no economic payoff to the society that sends off the starship. It can be done, but the motivation for doing it will need to be either political or religious. (One mode has mobile societies develop to mine the Oort clouds, etc. and then get into a political disagreement with the home planet.)