If docker isn’t your thing, you can run multiple virtual machines under KVM and make each service its own separate machine. It really doesn’t take much hardware to get started. If you meant doing system backups, you could always start with building a simple NAS from an old desktop machine, then run a cron job with rdiff-backup to make daily incremental copies of your other servers.
The point is, don’t let old hardware hold you back. I just moved my web servers off of some 20-year-old rack servers earlier this year, it really doesn’t take a lot. Just grab any machine you have laying around and get started. The practice will teach you a lot, and you can then figure out if you need a better system.