The No Man's Sky Artifact Exchange is located in the Colossal Archives.
Before you can trade in your ancient relics (which obviously belong to a museum) for rarer items in No Man's Sky, you must first locate a colossal archive. I have a full guide on how to do it here, but to sum it up really quickly in this article: take the cartographer's trade planetary maps and place them all in space until a "planetary archive" appears. Make your way to this waypoint and you will find a colossal archive, where an exchange of artifacts will await you.
There are usually a few of these terminals scattered around the archive, but the easiest to find is just outside the airstrip near the vendors on the upper level. It looks like a giant safe (shiny locks sort of give it away), and it will say "Artifact Exchange" when you go to interact with it.
You will need artifacts on hand if you plan on making a trade. In No Man's Sky, these are ancient biological samples, fossil samples or lost artifacts from ancient ruins, bone sites, etc. As long as you have one in your inventory, you can exchange it at the Artifact Exchange for a different artifact of equal or greater value.
Now, from what I've seen, whatever race the Colossal Records are a story for will determine which of the three has the best chance of "getting" to the next level. In my case, a Vy'keen archive turned all of my purple biological samples into rare and lost yellow artifacts. Two of them sold for almost 4 million credits, which was a huge increase from the 200000 I would have received for the two purple artifacts I traded.
I will update this guide once the community or I find out the exact science behind the Artifact Swap, but for now you can still use the Artifact Swap in No Man's Sky to return artifacts for potentially greater gains. Go out and prosper, travelers! For science, of course. . . .