So what would be needed? A box that could receive say 3 DME’s at a time and then an output to a moving map display?
With modern electronics could such a unit be the same size as a King DME 62/4?
Would you need 3 antennas?
Collins DME-42 from the early 80s: https://patentimages.storage.googleapis.com/28/38/08/1748970c1d2e48/US4642639.pdf
Basically time-multiplexing of up to three DME interrogations and signal processing on a single RF transceiver. I would bet that Trig/Garmin could start from their TXP line and implement the software more or less directly according to the patent specification … Can you imagine a device the size of Trig’s compact TT21/22 transponders giving you a DME-DME-DME multilaterated position?
Fly in the ointment might be the reduced availability of enough DMEs receivable at lower levels.
Bathman wrote:
A box that could receive say 3 DME’s at a time and then an output to a moving map display?
2 DMEs are enough. That does give two potential positions, but if they are at a sufficient distance apart and you know the initial position of the aircraft only one of the two positions is possible given how far the aircraft can move between updates. (I suppose a DME/DME RNAV box always chooses the two DMEs that way.)
A receiver could be designed to use the ADS-B UAT ground stations and determine position. The UAT frames are synced to the same time reference. In the US, there a more than 650 ADS-B ground stations. Europe could get on the band wagon and use UAT ground stations also to send FISB products to aircraft, so it would serve two purposes.
For the DIY DME, a source of inspiration could be the Mode C DIY tranponder posted here: https://www.euroga.org/forums/maintenance-avionics/6851-build-your-own-avionics/post/124813#124813
In fact you need only 1 DME to get position!
How does that work?
Ask someone who does satellite launches 
It has been a standard technique since the 1960s. You end up with 100s or 1000s of simultaneous equations and you solve them continuously. The solution is constrained by various factors; in our case it is aircraft performance and in the satellite case it is that plus orbital mechanics. You do need to initialise the system appropriately.
I was told by a Concorde pilot in 2003 (I was on one of the last flights) that they use a single DME for their INS corrections. I already knew of the single DME method so I didn’t question him (also everything was rushed) and prob99 he would not have known the mathematics involved. But he did specifically say it was 1 × DME because I mentioned the standard 2x DME method. It does take a fair bit of computation which in the 1960s was a fair bit of hardware, whereas today a $3 ARM32 chip can easily do.
How does that work?
You’re not going to get away with that. How is it done? Doppler?
The clue is in “multiple simultaneous equations” 
Peter wrote:
The clue is in “multiple simultaneous equations”
AKA rocket science
INS is amazing stuff. A gimbaled INS can align itself when stationary, you only have to enter the longitude. When moving in a straight line, I would think the velocity component in N-S is always known from the INS without any alignment. You will always be able to move straight E-W. Don’t know how the math becomes (except terrible), but an INS is an analogue computer that solves lots of equations all by itself. But the errors adds up over time, and it needs to be initialized and corrected.
Even if you are tracking a single DME beacon, the solutions are constrained by your heading (from the fluxgate magnetometer), your TAS (you have IAS + OAT + pressure altitude) and obviously where you think you have just been.
In reality any implementation has a database of DMEs and it tunes them as it moves along. This is necessary as they move out of range. The tuning of a DME box is instant, so you need only one. And if you are tuning the DME, you may as well use any beacon that’s in range.
The problem, as always, is that the volumes are low, and GPS is extremely reliable and precise, so few people are bothered. And there are solutions involving FOG INS combined with GPS – exactly what is done in missiles etc.
