Since you’re asking in a SuiteCRM forum, let me recommend SuiteCRM
You can host SuiteCRM in-house, or on a public server (old-style web hosting, or some modern cloud service, VM-based or container-based). Either way, you have full ownership of your full code and data.
If you host in-house, a separate decision is if you will allow access from the Internet, or just internally (LAN). But I would say there are many benefits of making it available online (with the customary security measures). This is definitely the most common scenario.
I would certainly recommend SuiteCRM as your software, but I would not recommend hosting it yourself, in-house. Hosting websites brings into play a whole gamut of things that will just drain your time and energy … responsibilities that you don’t need at the time of doing a CRM conversion.
With the low cost of hosting these days, you are much better using a hosting company and letting them worry about up-time, security, updating the underlying software (PHP, MySQL, etc.), creating your backups, etc. (Up-time and security should be major ongoing selling points to your boss, btw).
Let hosting companies focus on what they do best and you focus on installing and configuring SuiteCRM to fit your company’s requirements. Any CRM migration should not be taken lightly!