Can an employer require me to sign a non-compete for 3 years with a similar company doing similar work?

Your employer may require you to sign anything as a prerequisite for your continued employment. Whether it is an enforceable contract is a very different question. If you would like to discuss any issues further, please feel free to contact my office.  The link to my contact information is below.  Thank you. The foregoing is general information only, not specific legal advice. No attorney/client relation has been created or should be implied.

Generally speaking, California employment contract provisions that prohibit an employee from working for another employer after completion of his employment or imposing a penalty on the employee are invalid. 

However, there are some exceptions, including the employer's ability to enforce covenants that prohibit an employee from disclosing trade secrets and covenants that prohibit an employee from working for a competitor while working for the employer.

Based on your question, the non-compete might prevent you from working for a competitor while working for your employer.  However, once you are no longer working for the company, you are free to work elsewhere so long as you don't disclose any trade secrets. 

A business law attorney can help you determine whether any work you are doing right now qualifies as trade secrets.