Nowadays, "server" is a veeery broad term. Almost everything is virtualized: small services run in containers, medium sized services run in virtual machines, massive services run on clusters. Hardware is not very relevant and can be added if necessary.
For home needs, the focus is on storage: backups, sharing, streaming media locally. None of these are too demanding and low-power NAS units are more than enough. I've been very happy with Synology units, but I'd recommend the Plus range for anything that's not absolutely basic. Good enough for even small & medium business needs.
If you're more technical, you might want to look up "home lab": people setting up a small datacenter-like configurations at home, to learn / experiment or even use for productivity. This sounds more like what you're looking for. The ideal starting point nowadays for a home lab is one of the Tiny Mini Micro (also known as "1 liter PCs") units from HP / Dell / Lenovo (more details
here). They pack massive performance / watt, they are really cheap and plenty on eBay, upgradeable, take very small space and they are really quiet (especially compared to a typical datacenter rackable "server"). Installing a free software like Proxmox will allow you to create virtual machines or merge multiple physical units under a single virtual server. For example, a typical unit I use in my home lab is Core i7 (4C8T), 16GB RAM, dual SSDs, costs about £100 and uses 11W of power at idle and about 40W at peak load; for processing needs, this is more than most people need at home.
If you're doing something exotic (like 4k video editing over the network, etc) let us know, there are solutions for that as well. But cheap decomissioned servers rarely make sense at home, they're too big, noisy and power-hungry for that.