Be careful with "cheapest" -- you get what you pay for. The safest option probably would be to use large cloud providers like AWS, DigitalOcean, OVH, etc. and simply provision a new server whenever you want to launch a new blog. You can get a small server for around $5, which is relatively cheap but does add up when you have several PBNs, plus you'll have server maintenance.
There are PBN hosting providers that do all of this for you, of course, including regular backups. While perhaps not the cheapest route, it's certainly the most convenient and it is usually possible to find affordable options within your budget.