One of the Federation Cruiser's ship-specific achievements is Master of Patience:
Use only the Artillery Beam to destroy an enemy ship while taking no hull damage.
What is the quickest/easiest/safest way to gain this achievement?
I strongly suggest doing it in early system (1-3) at encounter with asteroids. If you have a bit of luck most of the work will be done for you.
Don't worry - damage from asteroids did most of the damage for me and I earned the achievement.
I've found that this can easily be picked up in the first couple of sectors.
All you really need is 50 scrap to upgrade your shield subsystem two four bars (two shield "bubbles" when fully powered). In general encounters, ignore this upgrade and leave it unpowered until you find an enemy ship that cannot get through two shields. This generally involves them having one weapon which takes a single shield off (basic laser, heavy laser, anti-ship drone) and any number of beam weapons (which are useless against shields*).
Once you've found such an enemy, it's simply a case of moving power from your Burst Laser to your second shield layer - and waiting for your Artillery Beam to destroy them.
This is probably easier to pull off early, because it's less likely that you can reliably soak all enemy damage in later sectors, even with fully-upgraded shields.
*I don't know whether enemies can have beam weapons that do more than one damage per section, and hence could penetrate a single shield, but I've not seen them. And it's safe to assume they wouldn't appear early in any case.