You need your own domain.
If you want to send email to yourself@your_domain.com you obviously need the domain "your_domain.com". The domain has to exist on the Internet and have mail server record(s) defined for it with a mail server waiting to accept email.
If you say that a personalised email is something like yourself@someone_elses_domain.com, then those emails will go to whoever owns the domain "someone_elses_domain.com" — and they are not going to let you do what you want with it.
Ditto with websites — if you want something personalised then you have to get your own domain.
Bottom line - you need your own domain.
It is not that expensive and/or too much hassle.
Finally if you are asking can OS X Server manage and host these things (mail, websites) then the answer is definitely yes.