2 TB is going to be difficult - I assume you already know it's going to be expensive :-)
In principle, an external drive (SSD or HDD) works like a USB stick or SD. You plug it in an access it. Theproblems that might show up:
- access speed. You'll definitely want a high speed USB port both on your PCs and the external drive (no matter whether stick, SD card or external drive)
- power consumption. Goes, for a part, together with access speed. Those large devices, especially if they're mechanical (HDD) need more power than at least USB 2.x ports can provide. A cheap and unreliable workaround was to provide them with dual USB connectors, which really is not a solution
Another option might be a cloud based (or otherwise online accessible) storage system. For example, at least in theory, my NAS at home comes with a software that lets me access my NAS' contents from anywhere (haven't tried that yet, but plan to try it for my music collection). Or yu could rent online storage space from any number of cloud service providers.
If online is not an option for you, I'd suggest you get an external SSD with a USB C connection.