It all depends on a couple of things, not the least of which what, exactly, you define as "too slow." And that is going to be answerable only by trying it out. One of the other things it depends on is how fast your network is.
Modern networking is at the gigabit level, but it is still common to see routers and switches, especially those provided by your ISP to be 100mb (or 1/10th the speed of gigabit). Needless to say I would recommend you have gigabit networking. And if you are relying on your ISP-provided router for your network, perhaps you will need an inexpensive gigabit switch to connect all your computers with one port left over to then plug into your ISP-provided router.
Another consideration is what, exactly, you keep on the external drive. If it is your user profile I would say no. If it is your documents folder, probably.
Again it depends on your personal usage patterns and how much data you are pushing across the wire and how fast you need it to be.
I have a NAS. I use it for backup (Time Machine & Dropbox) and as a centralized location for music, TV & movies (Plex) but don't use it as a location for keeping files I am currently working on (Dropbox is there just as another backup location). But people work with documents (Word, Excel, etc.) across a GB network all the time, so it is completely doable.
But it depends heavily on your use case. So find a place with a liberal return policy (Costco?) and give it a try.