Can I store the images of a Lightroom catalogue on a cloud or network drive?

The Catalog has, as a technical restriction, to be used from a location in the local computer...

so, internal or external attached storage, but not a network or a cloud location.

You still can SYNC / COPY instances of the Catalog file between local locations on different computers, via a Cloud based intermediate copy, so as to keep these all up to date with your latest work. For example, using Dropbox.

That replication would need to be happen at times when LR did not have that local instance of your Catalog, in use.

The restrictions are looser, for where the image files can live. You can certainly keep them in a folder on a network based drive. You can also map that network folder and have it emulate a local drive volume.

And while a cloud based location will not (I think) work directly; again, you could in principle maintain a local file store on each computer, and then keep those up to date with each other (through sync/replicate of a cloud based intermediate copy, again using e.g. Dropbox).

RP

The catalog cannot be stored on a network drive. I don't know anything about Apple's iCloud System since I'm a lifelong Windows user. But a lot of users have had success storing their catalog in a Dropbox folder and accessing that folder from different computers. That does not mean that it can be accessed simultaneously from different computers. You must take care to ensure that the catalog has had ample time to completely update after closing on one computer before opening on another or it is extremely easy  for the catalog to become corrupted.

It all depends on what exactly you mean by "store".

For Lightroom to use the catalog, the catalog must be on a local drive, internal or external; no network drives, no cloud storage.

While Lightroom is not using the catalog, you can store (like for backup purposes) the catalog on a network or cloud storage. You just need to remember to copy/sync it back to the local drive before you use it.

For Lightroom to use the original images, they must be on a local (internal or external) drive or a network drive; no cloud storage allowed. For backup purposes you could use a cloud service, but LR will not see the cloud service.

The easiest (not prettiest) way to work on your photos using multiple computers is to put the catalog and images on an external drive and move the drive from computer to computer as needed.