How are you planing on finishing the exposed wood? Paint? Stain?

This sounds past your skill level.

I plan to paint the cabinet. I don't see that the body of the cabinet will be difficult to buid. I just dont know how to build the doors, how to attach the mirrors to the doors, and where to place the hinges. If necessary I will have a glass establishment glue the mirrors to the doors if that's the right way to go. Just need some guidance on the doors. I'd like to give it a shot.

I'm sorry I missed this when you first posted. I just finished a cabinet like you've described.

The door installation is not trivial. It is, however, doable provided you're willing to obsess a bit over the measurements and go slowly. It follows that you should not make anyone you care about go without a medicine cabinet while you build one of these things.

The cabinet box is dead simple, though it, the doors, and the mirrors have to be pretty square. I found that drawing up plans in a CAD program, like Sketchup, is extremely helpful. Unless you are much more comfortable than I working in reciprocal powers of two, you will find it useful to switch from inches to millimeters when planning your hinge installation.

I really think the cleanest door installation is with 1/2" MDF doors and Mepla Tri-view medicine cabinet hinges (easily found by googling). These European-style hinges attach to the inside surfaces of the cabinet top and bottom (rather than to the cabinet sides), so you can mount them inboard, so that the doors swing properly. Since the cabinet mounts attach with small screws (which you may have to reposition slightly to get your doors to line up correctly) the cabinet frame should be decent wood, not MDF. At least the top and bottom of the frame should be wood.

The hinges are designed for 3/4" doors but you can install them in 1/2" material if you're careful. It doesn't matter if you break through the front of the door a little with the point of your Forstner bit when you're drilling the mounting hole because you're going to cover the front with a mirror (later).

MDF may be better for the doors than plywood because it tends to be and stay flatter. It's heavy, however, and I would definitely think again about using a lightweight plywood (Sandeply?) were I to build the doors again. You can get water resistant MDF, which is what I used. I think 1/2" is about as thick as you want to go due the the weight of the material (plus that of the mirror) and the thickness of the door plus mirror plus mirror-mounting material.

The mirrors will mount with special mirror-mounting tape and mirror-mounting mastic. I used lots of each, and they were expensive. Home Depot has them. I looked without success for some kind of mechanical mount (e.g., some kind of nut-and-bolt) that would be both secure and (to my mind) unobtrusive. It would be good to have an attachment system that permits replacing mirrors easily. The only way I know to remove a mirror attached to a door with tape and mastic is to dribble acetone out of a plastic squeeze bottle between the mirror and the door, which will almost certainly wreck the finish on the door and partially remove any veneer. With that in mind, I've finished my cabinet with Waterlox, which I am happy to report is every bit as easy to repair as they say it is.

An option to consider is to buy replacement doors at big box or specialty stores. Then remove the center panel from the back. Finish them to your liking, then have the glass shop add the glass. I'd recommend using "glazing points" and not glue them in.

If you need to make the door flush with the wall, go for Euro hinges, AKA cup hinges. They have 3 way adjustments. The only tricky part is drilling the 35mm hole in the back of the stile. But if you purchase pre-made doors, you might be able to get them already drilled and ready.

Just another option.

Here is a link that might be useful: custom doors

The 'special hinges' are often just short sections of piano hinge attached appropriately to the carcase and the back of the door.

Many thanks to all for your help. It is much appreciated.

Go to some stores 9especialy if there are any 'bathroom show rooms' around and not how the cabinets are put together.

You may see one you like, or at least you can get ideas about how to build what you want.