Yes, if you don't mind getting a little hands-on, you certainly can. It's possible to save money by buying an internal hard drive or SSD and a suitable enclosure and putting it together yourself; it'll take five minutes, tops. For the best performance, go for an NVMe SSD and enclosure, but bear in mind that to get top speeds out of it you'll need to plug it into a USB 3.1 Gen 2 port. For less speed at a better price, seek out a SATA SSD and a compatible enclosure; it's also easy to find hard disks and enclosures for maximum storage at the lowest prices.
Note that you won't benefit from any useful automated backup software that you might get with a ready-made external drive, so if that matters to you, you'll either need to source your own or settle for one of the options above.
It's also worth knowing that if an external drive fails on you, it's possible that the fault's in the enclosure rather than the drive. So if your external drive dies, it's worth finding a cheap enclosure and swapping the drive into that to see if it comes back to life; it's worked for us before!