Every orange quest is in two parts: first some solo quests, then a raid (12-man) dungeon. This part is not meant to be soloed.
You have 4 orange quest, so you have to do them to be able to enter these 4 dungeons - in difficulty order: Skybreak Spire (Raven King), Temple of Eluvium (Hive Queen), Nightfall Sanctuary (Grand Celestial Emperor, the hardest raid for now), and Scarlet Conservatory (Scarlet Empress, should be ingame in the next patch).
You can check the minimal gear requirements by pressing Ctrl+J
I suggest you to do the solo part of these quest, especially the last one (west of Imperial Bazaar, Dasari Palace) since they will give you some xp and stuff.