The chance to get this is not really astronomically low. Your dodge chance to complete this achievement would be 45%, 35% from the engines and another 10% from your pilot as long he gained no levels. This leaves 55% chance the attack hits. 0.55*0.55*0.55*0.55*0.55 = 0.05 = 5%.
You need to have a active pilot. Since pilots level up and increase the dodge rating you should try to get the autopilot system or have only a active pilot when the opponent is using missiles.
Obviously you should not build defense drones since a lower amount of missiles have the chance to be dodged as the drones shooting them happens first. Also it does not say you have to man
By upgrading the engines as fast as possible you increase the amount of possibilities this could happen so effectively raises the chance over time.
You should try to find ships that fire many bursts of low damage. Also a hull repair drone helps you last longer while trying to get this achievement.
Further more there is a possible bug with this achievement.
Taken from FTL wikia:
As of Steam Version 1.03.3 for Windows, I was unable to get this
achievement without also upgrading the Piloting Sub-System to level 3.
This doesn't seem like it should be necessary, but I let the game run
for roughly 5 hours against an opponent with a Level 3 Burst Laser (5
shots), watched combat for about 30 minutes and witnessed 5
non-dodged shots in a row many times, assumed many others happened
over the course of 4 to 5 hours while I was not watching, and could
not get this achievement. When I upgraded the Piloting Sub-System to
level 3, I got the achievement a few minutes later. This could very
well be a bug, but if you are having difficulty, you may want to try
upgrading the Piloting Sub-System.