If you have a basement with masonry walls, you already have two walls of your vault in a corner. When I built my house, I set rebar in the concrete floor for the inside walls, but you can drill and place rebar with "Setcrete" into the existing floor. The easiest way is to lay 8"x 16" concrete block with vertical rebar @ 8" o/c( one in each cell of the blocks, and wire ladder reinforcement in every other course( every 16"). In an existing home, the ceiling will be much harder than mine, where I could pour concrete without the obstruction of the floor above. You will have to evaluate your own situation and decide between 3 choices. If you want a concrete ceiling, you will have to build a form, using either 4x4 or 2x4 and 1x4 cross bracing. I used 2x4 @ 16 o/c, because I was going to have a use for them during the rest of the construction. You will have to "worry" 3/4" plywood overhead during the placement of the posts, or place them over the top of the walls by leaving the last course of block until after setting the form. You will need to think backward with the idea of being able to wreck( remove) the form. You don't want to have to resort to chisels. Be sure to run conduit for an overhead light and a convenience outlet for a dehumidifier. While you are laying the walls, make arrangements for a drain, so you won't have to empty the dehumidifier manually( it will run over otherwise). You can drill anchors into the existing walls, to tie the new walls in. Since you will be limited for "headroom", you will need to use short sections of vertical rebar and tie( wire) them together as you go up, and fill the cells in the block course by course as you go up. People that sell gun safes also sell vault doors, so you should talk to one of them in the beginning of the project to find the dimensions you need for the opening. Someone above recommended a 9' ceiling. You are not likely to achieve this, you will be lucky to get 7 1/2'( 6 1/2-7 is more likely). You will have to pour the concrete for the ceiling over the top of the wall, so you should orient the vault so that the joists of the floor above run with the short side, to make pouring the concrete a little easier.
A second option is to anchor angle sections to the walls and fabricate a steel ceiling, from what ever is available to you. I have seen them built from PSP( pierced steel planks- airfield matting) and can understand using bars or plates. The comments on lights, door opening dimensions and drain still apply.
A third option is to just make the walls run to the joists above and place a ceiling in the room outside and one inside the vault, hiding the lack of a secure ceiling. The other comments apply here too.
Are we confused yet?