The red x means you do not meet the level requirements for the gear, as in, you do not have any class that is of a high enough level. A green x means you do have the appropriate class at the appropriate level to equip the gear, but you currently have a different class active. Which level and class you need to equip a piece of gear is right there in the gear description, which I suggest reading carefully in the future.
Damage to your gear on the other hand is shown by the left one of the two small bars on the side of your equipped gear's icon, as well as the colour of the dots that represent your equipped gear in the bag slots bar. When gear starts to get damaged, the dot and bar is green, and the bar will go down. When it turns red, that means it is now too broken to give you its stats benefits. Repair it to get it back to working condition.
NPCs that repair your gear for you are called Menders, but at camps outside the city, you can often also repair at an armour merchant. Repair Gear is separate a menu point when you talk to them. Repairing at an NPC costs only gil, and they will repair all gear that is not at 100% indiscriminately.
You can also repair your gear yourself, that is what the 'Repair' menu point in the right-click menu of your gear icon is for. Repairing your gear yourself requires a piece of Dark Matter, which you can either buy from NPCs (normally found under 'Tools') or gain from Mining, as well as a high enough level in the crafting job as indicated in the gear description (f.ex. lvl 1 Leatherworker).