How do I keep ships repaired in the field in Endless Space?

From the FAQ thread in the official forum:

How do I repair damaged ships?

Ship repair themselves a certain percentage each turn they are in a friendly system, Hero Abilities and Battle Actions can also be used to repair. Installed ship repair modules do repairs both in combat and on the Galaxy Map. Specific tech can increase ship repair rate.

They do repair themselves extremely slowly outside of battle, as far as I have observed. I tend to use the repair nano bot card during battle a lot to help keep my fleets at full health.

Ships repair themselves each turn - there are different modificators on repair:

• own system means highest repair rate
• repair technology provides fleet repair per turn (galaxy map)
• repair technology provides ship and/or fleet repair per combat cycle
• outside of combat repair techniques even work on different fleets. Meaning: You could have a fleet of fast repair/support ships in a different fleet and warp 'em in after your fight ;)