Yes, you can store and use a Lightroom catalog on most external drives.

However, you can only store, but not open, a catalog while it is on a network storage drive or NAS. Adobe does not allow this due to the fact that multiple computers could be accessing the same catalog at once. If you use a NAS for storage instead of an ordinary hard drive, you can store your actual photos on the NAS, of course, and you can back up your Catalog file (.LRCAT) to the network, however, to actually work on the catalog you’ll need to transfer it to a local computer hard drive, or an “ordinary” external drive. Simple 1TB SSD drives work very well.

Read more about How To Create A Lightroom Catalog and Import Photos