The OLD old school version.. How far back could you have made it work?
As a rough guide, to get something that's recognizably DF, I think you'd need:
* a true 32 bit microprocessor
* built-in or add-on math processor
* a MMU with proper 32-bit virtual memory support
* several tens of MB of memory (RAM)
* a hard disk drive with several tens of MB of space free after the OS
* a directly connected display capable of around 640 x 480 with 256 colors
* ordinary US wall plug power
* no external cooling requirements
* readily available compilers
Off the top of my head, an Apple Macintosh IIx was one of the first "out of the box" personal computer to meet the above criteria; available in 1988 loaded out for around $10k; as a bonus, it was used in a variety of university settings back then, and so possibly familiar to an Alt-Toady from his previous academic life, or even acquired surplus a few years later.
A tricked-out Amiga 2000 might have done the job around the same time, but IIRC wasn't available stock with the right accessories until the Amiga 3000, which came out a few years later in 1990.
To look at things from a different angle, you really want a Motorola MC68030; you might be able to try with a MC68020, but I'm not sure you can go much earlier. At the time, the Intel-side offerings were much weaker on the MMU side; you could probably manage something with an Intel 80386 + 80387 + supporting chips; but while back in the day a "Compaq 386" was cheaper and better for office and some engineering uses, the university programming departments I was familiar with used the Apple & Amiga 68k-based lines for fairly clear reasons.
So, very late '80s, almost certainly; although embark sizes would be rather smaller than we have today, and it would probably not be thought of as a "real time" game; think more like a Roguelike, at low-single-digit FPS.
Note that of the philosophical roots of DF, Angband itself only dates back to around 1990; the much simpler UMoria was widely distributed around 1985. So, again, it's unlikely much before the very late '80s, and early '90s is more likely for a plausible time for someone to have the idea of a persistent, evolving surface world for a Roguelike that could be run on a desktop of the time.