Not through the cloud , as it will slow the process.
You can put the catalog and al the images to external hard drive and make sure all the images in the catalog refer to the images of external hard drive.
Now as this catalog is on external hard drive, you can take this hard drive anywhere and use the catalog on the machine where Lightroom is installed.
However you can't store catalogs on a network but you can store your photos on a network. Smart Previews let you edit your photos in Lightroom even when you are disconnected from the network or when your computer is disconnected from the drive that contains your photos.
Check this FAQ of Lightroom : Lightroom catalog FAQ
The other person would have to log into LrWeb using your account, but they could then do some basic adjustments, ratings, flags, share collections, copy and move items between collections. Those changes would then automatically flow into your catalogue.