Can someone that is on probation for domestic abuse and if they are depressed go in to the military?

Probably not, no. Depression alone will keep a person out.

The Army refuses to train (including firearms) people who are depressed.

If you were convicted or pled guilty, you would require a waiver to enlist in the armed forces. Receiving one will depend on the severity of the offense, your behavior afterwards, how long ago the offense happened, and the needs of the military vs your skill set. Each branch has its own rules on what offenses can be waived and what will disqualify you from service. So, long story short, it is a question that only your desired branch of the armed services can answer for you.

Most likely no you will not be able to go, you will have to get two waivers in fact one for your medical condition ( that will be hard) and one for the criminal offense.