Rent a VPS or buy an old desktop for hosting side projects?

If you're working in software, don't waste time on the hardware. It's gonna be a pain in the ass just to deal with IPV4 and DNS behind your university network or your home router – and likely to be against the terms of service of either one.

For random websites, just use Vercel or Netlify for free. For the other lower-level stuff (k8s and random backends) sign up for free educational credit from the various big clouds (e.g. https://aws.amazon.com/education/awseducate/ or https://azure.microsoft.com/en-us/free/students or https://cloud.google.com/edu/students?hl=en). Maybe cycle through them once a year so you can keep using them for free and also get experience with several clouds.

After uni, unless you're working for very large companies with their own data centers or specific industries who have to colocate their own servers (usually for compliance), you'll never be working with hardware anyway. It's all in the cloud, so you might as well learn that (there's a lot of management and auth/permissions overhead) instead of dealing with x86 hardware.

It depends on what you're aiming to learn. If you're interested in handling hardware and connectivity, hardware is your best bet. If you want to focus on programming, the VPS is likely your best option. They're relatively affordable, and you can learn how to host a project using one, allowing you to focus on programming.

Buying and maintaining your own equipment is a good learning experience in itself. But universities generally don't like it when you host things on their networks, so that should be considered as well.

I’d avoid the hardware. It’ll just slow you down or become a distraction… not to mention that everything takes longer. Want to wipe a VPS and start over? A few clicks. Want to take an image of the system? A few clicks. With your own hardware you’ll waste an hour just trying to get the dynamic IP software working, only to find later that certain ports are blocked by your ISP. Just no.