good old x201 here (i5-720m iirc), 8GB ram, sata ssd. Debian stable. No DE, just stumpWM. Not watching 4k youtube videos but runs fairly well for a 13 years old machine.

If you are in France, or around Europe and don't mind sending your Pi via mail to them, Faimaison and Tetaneutral do propose small computers hosting in their datacenter racks, Pi type included, but also NUCs, respectively for 24€/month (bit expensive but small structure compared to Tetaneutral edit: it's ~15€/month nowadays) and 5 to 10€ / month. That's just an example. Generally you'll get one IPv4 and one /56 or /64 IPv6 prefix.

You might want to look near your location if there is a LUG, non-profit ISP, or non-profit colocation proposing the same kind of services. You may even meet some nice people! But it's definitely doable at least in Europe.