Want PhantomBot running remotely all the time? Get an affordable Virtual Private Server (VPS).
Vultr - US-based, but also has locations in Europe. Cheap and good quality.
DigitalOcean - US-based, but also has locations in Europe and Canada. Fairly cheap for their lower tier options, gets progressively more expensive.
Scaleway - Europe-based, but is very cheap for what you get.
ovh - Europe-based, but doesn’t have many locations (does have a Canadian based location).
Linode - US-based, but has a lot of locations for a good price.
Amazon Web Services - Free for a year (with certain limits), but very pricey after that.
Azure - Microsoft-owned. Not on the cheap end, however.
Compute Engine - IaaS | Compute Engine | Google Cloud - Similar to Amazon Web Services, but owned by Google.
http://lowendbox.com/ - A curator for lower specced servers.
You can always self-host on your own hardware. A Raspberry Pi 2 B will be more than sufficient for small to medium sized bots.
For bigger bots, you can build your own server PC for usage, or buy a rack server.
Any modern hardware should work 100% fine.
No. There is no good free VPS hoster, outside of persuading somebody to host for you, which is incredibly unlikely.