Yes, it certainly is possible to become addicted to things other than drugs or alcohol. An addiction is something that almost consumes a person's time and energy, and that they can't break out of on their own.
Occasionally, I get letters from people who spend all their time on the Internet, or on constant exercise, gambling, pornography, or some other bad habit they don't seem to be able to break, although they usually know it's harmful and could even destroy them. Psychologists I've talked to say these activities can become just as addictive as a dangerous drug -- and as destructive. I'm not qualified to know when a bad habit crosses the line and becomes a true addiction, but I know it can happen.
I'm curious, however, why you've asked this question. Has some habit taken root in your life so deeply that you fear you can't break free? If so, face it honestly, and then ask God to help you overcome it. It won't be easy, and it may require the help of others, but don't let pride or anything else keep you from seeking help.
Whatever your situation, make it your goal to live for Christ and seek His will for your life. God loves you, and when we commit our lives to Christ, He comes to live within us and help us by His Holy Spirit. Make your commitment to Christ today. Then let the Psalmist's pledge become yours: "I will be careful to lead a blameless life.... I will walk in my house with blameless heart" (Psalm 101:2).