You can have a hosted e-mail system with your own domain, many service providers will do this. They would have the storage, archive, etc. on their servers. Access would be a web client, e-mail client of your choice, or a bridge demon. Option 2 is a hosted vertual server, so your self hosting but renting a VM server on a providers data center, they take care of the hardware, power, backup (sometimes depending on service), etc. You would set up your OS and e-mail services.