How far apart should I build my cities in Civilization 6?

I know in many (possibly all?) previous Civilization games, cities could only grow to a certain size, and usually you wanted to place your cities far enough apart that they'd all be able to grow without overlapping.

How far apart should I place my cities in Civilization 6 to make sure they don't overlap?

If you want exactly zero overlap and maximum space utilization - place them within 6 empty spaces of each other (each city has workable surrounding area of 3 tiles away from City Center).
It is though reasonable quite often to space them 4-5 tiles away from each other as cities don't usually grow that big.

See this answer to a related Civ V question for visual aid.

With the new housing mechanic cities just can't grow that big that fast. Also border expansion is way slower than in civ5. Unless you invest tons of money into tiles there is no chance you will ever work the outmost tiles before industrial era or so. Thus I recommend against spacing out too far. Make sure you cover the important ressouces within two tiles of your city centers, that should be a priority early on. Spacing might be a problem, though, late game when you want to build lots of districts and wonders.