Fallout: New Vegas

Very easy to get, Once you've done the speech challenges to entice Elijah to come down to the vault, make a save so that you can come back and get the "Cash Out" achievement.
You need to open the vault door and go left up the small steps to ths area you came in through, you can wait if you want to see Elijah turn off the force field and then come through the doors into the area infront of the vault, but you dont have to.. Now just run back through the areas with the hologram security bots, Dont worry about them seeing you, just keep running through the maintenance shaft until you come back to the lift,
As in the video, just jump down from the level near the alarm terminal to save time..
The achievement should pop before you get to the lift.

If the Youtube video doesn't work, I also uploaded the video here:
http://game-play.tv/video/Y9KURD246DDY/Fallout-New-Vegas-Dea...

Before you actually trap Elijah in the vault, you have to convince him to come down. A repair, science, or explosives of 75 will do, and if not, a lockpick or sneak of 75 will make him come down after you. (A little extra help, the lockpick and sneak challenges can be activated by telling Elijah that you will take the treasure for yourself) Once you can move again after he says he's coming down himself, run out the vault door and up the steps to the left, but do not go through the doorway to where you entered from, just wait in the dark there where you can still see the vault door and the bubble around it.

Once he comes down, go into crouch mode (if you aren't already) and wait for him to reach the area where the computer and vault door are. Once he reaches that area, start moving (still crouching) and remain undetected until you reach the area that he entered through. If you are having problems staying undetected, there is a gun case in the same room as the terminal that has a chance of having up to six stealth boys in it. Other then that, you can read any La Fantomas that you have, make sure to keep the light on your pip-boy off, or try wearing the assassin's suit, which is located in the Medical district of the Villa.

Once you are in the room he came from, you are home free. He will activate the blue holo-doors and activate your collar, and you will have about 30 - 40 seconds to leave through the elevator before you explode, which isn't a problem. Once you leave, not only do you get an achievement, you get a challenge with the same name that gives you extra xp. If you have any problems with this guide, feel free to comment before you give a negative vote.

This achievement is missable!
Save your game immediately after talking to Father Elijah (through a video screen) in the Vault. I suggest saving here because you can easily screw this up and you will want to reload your save and not have to sit through dialogue again!

To get this achievement you have to escape the vault undetected.. this means Father Elijah has to come down the elevator to the vault, open the barriers, and start walking toward the main doorway to the vault. To sneak past him, you will want a high sneak skill or a stealth boy. I found one stealth boy while searching around outside the Sierra Madre (I do not remember the exact location, but I do know that it was on the second floor of an area and inside the room were like five desks filled with sierra madre chips and one stealth boy).

It is very easy to get caught by Father Elijah, even if you have a stealth boy. So this is what I did to lock him in: I waited inside of the vault with the door closed for about a minute, then I took my stealth boy, opened up the vault and closed the door behind me and waited (while sneaking) along the right side of the dome surrounding the entrance to the vault. Father Elijah, if he does not detect you, will open the barriers guarding the exit, and leave them open while he walks toward the vault. If he detects you, the turrets will come alive and he will close the barriers and yell, "did you think you can really steal from me?" (if he does this, load your save and try again!) If you are undetected, sneak through the now open walkways toward the elevator and your achievement should pop, because Father Elijah will trap himself down in the vault after reading the safeguarded file on the terminal.

Good Luck! If you would like to downvote, please let me know so that I may improve this solution. Please leave any additional comments and I'd be happy to add it to the solution to help others =)

Its a simple 1 once you exit the vault chamber with all the gold hoof it to the left back the way you came in make sure the old coot doesn't see you run back that way once you reach the shielded room the cheevo should pop not much strategy is needed.

Vid below for ya!


Good luck hunters

Credits to Jhaller2 for the video

Dont get me wrong the above two guides work but personally i was having trouble avoiding elijahs's gaze due to a spectacularly low sneak skill.

Solution,
-Convince Elijah to come down to meet you face to face, this requires a repair, science or explosives of 75.
[GOOD SAVE point if you want multiple ending achievements]
-Once he is on his way leave the vault into the middle of three rooms, (to your left is the way to run out and to the right is the room elijah comes from.
[GOOD SAVE POINT]
-Enter the right hand side of this middle room and follow it as far away from the vault door as you can.
-Take your time here if you've done the rest quickly you now have plenty of time for Elijah to reach the door and realise you're not there.
-Slowly follow thsi walkway round (sneaking).
-As soon as you're through the doorway you've made it.
-A short conversation with Elijah follows and entrances back to the vault from where you are close.
-After this there is one door left you can use follow this corridor take the lift and enjoy the G.
(mine popped during loadscreen after lift)

First guide so more than open to suggestion and imrpovement, good luck.

Took a couple tries to figure it out, but it is actually very easy. As soon as your conversation ends with Elijah just sneak out of the vault and run like a bat out of hell back the way you came. Elijah should be preoccupied long enough for you to get out of range of him and his turrets. In fact I was able to get to the elevator just as he noticed I was gone and started the security alarm.

I got the achievement without a stealthboy and acquired all gold bars using the very helpful video below. All credit goes to ExXuhLence.

After you get out if you have a home in the Novac apartments to store your gold bars. Closest place I found to store the very heavy gold bars and not have to walk too far while over-encumbered.

The perk long haul allows you to fast travel while over-encumbered.
requirements for long haul perk:
Level 12
Endurance 6
Barter 70

Both Cash Out & Safety Deposit Box can be obtained in one playthrough of dead money.

First save your game before you enter the vault!

Once inside the vault the door will seal shut. Dont worry, just use the terminal at the far end of the room. (note: the 2 at either side of the door disable security systems and whatnot...not important) 

Select the Vera file. (Vera Keyes does nothing that I know of) 
After reading it back out. DO NOT select the new file that appears. Once you back out Elijah appears on the monitor. You need to convince him to come down to the vault. This requires a Lev. 75 barter, repair, or science. Once you convince him to come down this is where you decide which achievement you want.

For Cash Out, wait just inside the vault door gun cocked, dont open the door until Elijah is in the area just outside the vault. (This may take a few tries to judge where he is on your compass) Once he's there open the door and kill him. Achievement popped after he died for me but I've heard rumors of it popping once the fight begins or when he enters the area.

Once he's dead escape, enjoy the ending, and reload your game when your ready.

For Safety Deposit Box, as soon as Elijah says he's on his way down and you can move, run out of the vault, go straight ahead up the stairs, and in the door on your left to the place where you entered this area. Run back the way you came through the maintenance tunnel and when you near the exit maintenance door Elijah will begin speaking. Wait a few seconds and the door will open. Run to the elevator and the achievement should pop once you get on.

After entering the Mardre's vault you need to convince Elijah to come down and meet with you - there are various speech related challanges you can pass whilst speaking to him on via the holo-screen. When Elijah say's he's coming down you need to quickly open the vault door and head towards the exit on the left side of the room before the force-field blocks the doorway - (I have the 'Travel Light' perk 10% faster movment) This best way i have found so far.... good luck guys!