There are 8 companions in total, two robotic and six humanoid. You can only have one of each type in your team at a time so if you want to recruit other companions you'll have to dismiss them first by talking to and parting ways with them. Make sure after you recruit them, you leave them on your team for a short while before dismissing them otherwise it may not register as them having become a companion. I would recommend at least leaving the area where you recruit them before dismissing them. If you're STILL having trouble unlocking the achievement keep each companion on your team for a good chunk of time, upwards of 5 minutes or so. Also, it seems that if you are on good terms with Mr. House you can send your companions there when you dismiss them so you only have one spot to go to change them out. For the players going for this during their Hardcore playthrough, don't worry if any of the companions die on you, you will still get the achievement when you recruit the last companion. This has to be done in one playthrough.
Robotic Companions
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ED-E - ED-E is in Primm inside of the Mojave Express building on a table just to the left as you enter. Before you can recruit it you will have to repair it though. You will need 35 repair and 55 science OR 65 repair OR 3 scrap metals, 2 sensor modules and 1 electrical part. If you need a small boost in repair there is a Fixin' Things magazine around the corner on the same room to boost it + 10. You will not be able to recruit it if Rex is currently on your team. Perk: Enhanced Sensors - enemies are easier to detect even if they are cloaked, you can also select cloaked enemies to target while in VATS.
Rex - You can recruit Rex into your team by speaking to "The King" in Freeside inside of the King's School of Impersonation. You will have to bribe the guard (either with caps or dialogue) in front of a locked door just to the left once inside. Eventually while speaking to him you'll have dialogue options related to his dog. Follow this path and he will allow the dog to join your team. You will not be able to recruit him if ED-E is currently on your team. Perk: Search and Mark - chems, firearms, and ammunition within a short distance are highlighted when you zoom in.
Humanoid Companions
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Arcade Gannon - Arcade Gannon is a medic located in Old Morman Fort in Freeside. His tent is located in the back right of the area. In order to recruit him you will need 75 speech or have the perk "Confirmed Bachelor". You'll get a "unique" dialog choice with him. He will not join you if you are aligned with the Legion. You will not be able to recruit him with any other Humanoid Companion already on your team. Perk: Better Healing - all forms of health recovery increased by 20%.
Boone - Boone is located in Novac at the top of the dino starting at 9pm. In order to recruit Boone you will have to complete the quest One for My Baby and have an Intelligence of at least 6. MAKE SURE THAT YOU HAVE HIM KILL THE CORRECT PERSON AND HAVE THE EVIDENCE TO PROVE IT TO MAKE SURE HE WILL NOT BE HOSTILE TO YOU. Talk to everyone that has a marker above their head in town to get evidence or the way to get evidence. When talking to him after the quest is complete ask him to come with you and when he asks for a reason use the intelligence option. You will not be able to recruit him with any other Humanoid Companion already on your team. Perk: Spotter - Highlight enemies when actively aimed at them (using a scope).
Cass - Cass is located in the Mojave Outpost south of Primm in a bar to the left as you enter the gated area with two buildings. When you first talk to her you won't be able to recruit her but must instead do the quest Heartache by the Number until you are tasked to find her. In order to recruit her you will need either 50 barter OR 75 barter OR 50 speech OR 75 speech (four separate paths). You will not be able to recruit her with any other Humanoid Companion already on your team. Perk: Whiskey Rose - No negative effects from consuming alcohol plus consuming whiskey give an increase in DT of 2 - 6 points depending on survival.
Lily - Lily is located in Jacobstown just in front and to the left of the large red building. In order to recruit her you will have to speak to the doctor inside the building in a room to the right as you enter. Discuss curing the schizophrenia in Nightkin and you will start a quest where one of the optional objectives is to recruit her. You will not be able to recruit her with any other Humanoid Companion already on your team. Perk: Stealth Girl - 200% increase in duration for stealth boy and extra 10% damage from sneak attack criticals.
Raul Tejada - Raul Tejada is the hardest to recruit. He is located on Black Mountain quite a bit east and slightly north of Goodspring in a prison surrounded by a lot of strong Nightkin and Super Mutants especially if you're a low level. As you head up the road you will be warned by another Super Mutant who will start a quest. After you finish the quest by killing Tabitha you will be able to free him. Make sure to read terminals to find clues on how. He will join if you ask. Perks: Regular Maintenance - slows decay of armor and weapons by 50%, Full Maintenance - slows decay of armor and weapons by 75% (obtained through dialog at end of an unmarked quest, either convince him that old people should retire and live long and useful lives or you will need 66 speech), Old Vaquero - His rate of fire with revolvers and lever action firearms is 33% faster than normal.
ALTERNATIVELY AND EASIER Try going all the way around the mountain from Neil's shack to the north, keeping to the high ground as much as possible. You'll pass near the Rangers Safe House or something like that. Shorty after that, there'll be a decent path you can go up the mountain on, and you'll be at the village there. Sneak around the village away from all the super mutants, and once you get to the path up to the radio tower, bum rush to the storage building, ignoring the nightkin. Once inside the building, fix Rhonda, tell it to follow you, then go back outside. Tabitha will thank you, then leave. All the nightkin will now be passive, and Raul can be gotten at your liesure. Check the unlocked terminal, read entry 6, then unlock his door with the other terminal. Tell him he's free to go, and when he complains about being all alone, tell him to come with you. You will not be able to recruit him with any other Humanoid Companion already on your team.
If you decide not to recruit him or can't right away or merely dismiss him, Raul goes back to his shack, which is just south of Nelis AFB. After you tell him to leave your party he says he'll go back to his shack, all alone in the middle of nowhere. It should pop up on your map.
Veronica - Veronica is found on the path from Novac to Freeside in the 188 Trading Post. When you talk to her she will ask what your opinion on the Brotherhood of steel is. The way that I got her to join was to reply that they are no threat as long as you don't have any good weapons and you will either be able to ask her to join or she will ask to join. Just stick to non-negative dialogue options and you should have no trouble recruiting her. She will not join you if you have made the Brotherhood of Steel your enemy. You will not be able to recruit her with any other Humanoid Companion already on your team. Perk: Scribe Assistant - Lets you craft workbench items through dialog with her.
Glitches(?)
People have reported getting the achievement with only 7 companions.
CounterInsurgnt reported that his followers would disappear if he tried to enter a place and a guy said his followers couldn't come with. Workaround seems to be telling your follower to wait outside before entering that place.
Some people have reported losing a follower (not killed or actually physically stopping following you, they just disappear through normal travel). Workaround seems to be going down the elevator in Vault 22. It seems that they might appear if you merely wait for a couple of hours (using the wait feature in-game). Similarly if you tell a companion to wait and then accidentally leave without telling them to follow you again, after a couple of days in-game time you will receive a message that they have returned to their post.
If you have anything to add or if I've made any mistakes, please point them out and I will update. THANKS TO EVERYONE WHO IS HELPING MAKE THIS A BETTER GUIDE, KEEP IT COMING!