Most Euro-hinges have x-y-z adjustments on the mounting side, so "exact" is not needed. But, that said, your old screw holes on the cabinet should be equidistant from the center line. Measure the distance between them, divide by two and mark, then run perpendicular line using a square from the front edge of the cabinet through that point?

Most euro hinges also have a small lip that helps with the back-to-front alignment; if yours have this, then the issue is top-to-bottom alignment. My preferred technique is to mount the hinge cups on the door, clip/attach the other piece of each hinge onto the door hinge-pieces, and then hold the door up to the cabinet (again, align the small front lip for front-to-back placement). This always gets me close enough to be well within the range of the x-y-z adjustment that Bob's nephew mentioned.

You were both correct, but we ran into another issue:

We had it all figured out and were about to start drilling the holes when my husband noticed that the original holes are drilled into a block of something, which is most likely not present in the rest of the area. You can see metal threads behind the holes from the original hinges, and nothing but air behind the holes for the shelf supports.

So what do we do now? I have been unable to find a hinge that matches the holes on both the door and the cabinet body; this one matches the door and I had thought that having the new holes far from the original ones would reduce the chance of having the holes rip out or otherwise damage the cabinet. But now we're realizing that the metal is not strong enough to support the hinges on its own (these were cheap-o to begin with so you know they did not put in anything extra just in case).

Is there any way to salvage the situation or have we hit the dead end that says "replace the whole thing"?

Sorry, but I can't quite picture what you're describing. Could you post some photos?
I suspect that adding/replacing the blocking will allow you to make use of what you have, but sometimes it's just easier to start over.

Try http://www.wwhardware.com They have about 40 pages in their catalog of euro hinge options, diagrams, dimensions, mounting systems, and carry multiple brands. They also have an 800 number that will answer questions. They sell these hinges in two parts, the door part and the cabinet part and you mix-and-match. So you can just order the mounting plates, saving even more. Even better, I have found the prices very good. I've got an order in for 24 hinges now, on its way to me today.

Thanks, Bob's Nephew, I'll check them out.

Salex, here's a picture - let me know if this helps.