I wanted to centralize the logs from my servers so that I can more easily access and process them. I found this tutorial for setting up centralized logging, but it didn't specify any hardware requirements.

Has anyone worked with something similar? Roughly what hardware specifications would be required for a setup with 5 servers that give a few lines of output every 10 minutes, but can rapidly shoot a few thousand on occasions.

Pretty much anything will work for only 5 servers; that's not a ton of traffic at all. Size it according to the OS you choose, and if it's for a corporate environment I'd suggest something long-term support like Ubuntu LTS or Rocky. If you're going to pay for support, SuSE Enterprise or RHEL. The only thing I'd watch for is disk space, but even that is trivial these days, since you'd have to wait a LONG time for five servers to fill up a 2TB (or more) disk, if that's all this server is used for.

Since you're going to do a centralized log server, be a bit forward thinking now and consider if you want to do server-monitoring with something like Nagios or Zabbix, which would also be a good fit for that box. All the other servers will already be whitelisted/through the firewalls so doing SNMP monitoring will be simple to implement.

That task is going to have a super low load on the system so just about any hardware would do, even an old laptop. So it comes down to operational costs, especially electricity.

SSDs are rated for a few years now and you can get various M.2 SSD HATs for the Raspberry Pi 5 these days, so that might be the cheapest yet still with good performance. That'll have more CPU and RAM and network capacity than you'll ever need for logging five servers even once you throw in extra monitoring software as suggested above.