When your character takes damage a little health bar will appear next to him showing some of it missing. If it appears, then you failed.
The easiest way to do this is by doing it with 3 another players. They can get shot but as long as you don't get damage you are good to go. The best way to avoid any damage is by getting into a mech (EXO-44 Walker Exosuit) or HAV. Bring backup mechs/HAVs! You can call in stratagems while inside the mech so don't leave it unless you really need to.
It doesn't matter if the mech gets damaged, it won't count towards the trophy only your characters body health counts. Also, if you get killed by pods falling from the sky you can still earn the trophy because pods are not counsidered as enemies.