You do not have to be an engineer to drive a car...
Salesforce is designed for building native apps or managing the platform without having programming knowledge just using the web interface and declarative application framework.
The Application Framework lets you customize existing applications or
build applications from scratch without writing any code.
You can even achieve Force.com Developer credential without having to write any code:
A candidate for this exam is not expected to be able to develop code
using the Apex programming language
Or just become an Force.com administrator. In this case you will have to learn applications, features and functions available to an end user, and the configuration and management options available to an administrator across
the Sales, Service, and Collaboration Clouds.
Take a look at the Force.com Fundamentals and learn more how to become an Salesforce professional developer or administrator.
So from my point of view yes, you can start without programming background. Just dive in an learn!