I’d say that single core performance and amount of RAM you have are the biggest issues with running anything on old hardware. Apparently, in theory, you could run even modern kernel with just 4MB of RAM (or even less, good luck finding an 32bit system with less than 4MB). I don’t think you could fit any kind of graphical environment on top of that, but for an SSH terminal or something else lightweight it would be enough.
However a modern browser will easily consume couple gigabytes of RAM and even a ‘lightweight’ desktop environment like XFCE will consume couple hundred MB’s without much going on. So it depends heavily on what you consider to be ‘old’.
The computer at garage (which I’m writing this with) is Thinkstation S20 I got for free from the office years ago is from 2011. 12GB of RAM, 4 core Xeon CPU and aftermarket SSD on SATA-bus and this thing can easily do everything I need for it in this use case. Browsing the web on how to fix whatever I’m working with at the garage, listen music from spotify, occasional youtube-video, signal and things lke that. Granted this was on a higher end when it was new, but maybe it gives some perspective on things.