Two possible options might be:
- "big number" classes, such as this one, which represent arbitrarily large numbers through mathematics on arrays of integers used to simulate an appropriate storage space.
- hierarchy; that is, using a tiered coordinate system possibly represented by two integers per component; the first integer represents the position of (say) a planetary system within a galaxy, and the second represents position within that system (et cetera)
Fundamentally these are very similar approaches, differing mainly in the interface they present over the multiple-integer abstraction.
A given game may choose one over the other depending on it's needs. For example, in a space game like No Man's Sky, you might only be able to "warp" between star systems. In this case there is a distinct transition phase that the game can use to smoothly transition from one coordinate space to another.
However, you might be able to smoothly and arbitrarily fly from space on to a planet surface, in which case you might have a harder time masking the coordinate system transition (it's doable but might be annoying depending on the rest of your game's implementation and how you deal with straddling cells or something). In that case you might prefer to represent coordinates using a "big number" class that simply let's you pretend you have a larger coordinate space to work in, at the minor cost of some performance now and then.