How can a company have its own Certificate Authority for internal use? It is possible for a company to deploy its own CA ? How would browsers accept the certificates ?

Supposing a company does not want to use the role of a commercial server like Windows Server or any other as a CA, how can a company deploy its own CA ?

Are there any reliable free projects that can help? 

Or the company needs to hire high-level experts, get its CA approved by top level CAs and then the browsers will accept its certificates.
This is done fairly often in companies, but any browser, client, or device which is to trust the internal CA needs to have the internal CA's root key added to its trust store. Browsers which haven't been set to trust the company's CA will not see its certificates as valid.

It's possible to run a CA using only openssl, but very painful. There are projects which make it significantly easier, such as FreeIPA.