I got sober without rehab so I know it's possible. I did spend 3 days in a hospital setting getting medically detoxed, but two of those days were a weekend so it was pretty easy not to tell my employer.
But you need to have a serious program, outpatient, therapy, counseling, AA or whatever.
If you've just been trying to get sober by yourself, like a lot of us try to do, it's very difficult.