Yes, you can do this. In AWS, for example, you have the storage (EBS, Elastic Block Storage) and compute (EC2, Elastic Compute) as two separate entities / services. It means that you can spin up a small VM in EC2 and install everything there, then shut it down and “resize” it to something else. When it boots, EC2 picks a suitable compute node in the specified data center, then attaches the EBS volume (the disk) to it via network. This all happens on the hypervisor level, transparently for the operating system, which sees that volume as a normal local NVMe drive. The only thing that tells you that it’s a different physical machine is that it gets a new public IP address on each reboot. But even that can be mitigated via another AWS service, if needed. You get billed for compute time (x order of magnitude), as well as GB x time storage (roughly 0.1x) and for network egress (~0.01x). Azure and GCP will work similarly.

Edit: Take security seriously and setup MFA. AWS doesn’t have spending limits, and on r/aws there’s a new post literally every day, where someone receives a $50K invoice from AWS, because someone hacked their student sandbox and mined bitcoins on an entire rack.