If your child support obligation is based on your making a higher amount of money than you really do, then you can request that this amount be modified. You can do this either by filing a motion with the court, or administratively, through the Oregon Department of Justice. You will need to provide proof of your income - usually, your four most recent pay stubs, and your most recent tax returns. More information is available at http://oregonchildsupport.gov/Pages/Index.aspx
That's the good news. Here's the bad news: Under Oregon law, child support is calculated by an algorithm that factors in the amount of time each parent has with each child, and the income of each parent. But each parent is presumed to be able to make at least minimum wage, full-time. At the present time, that's $1,525 per month, before taxes. If you make less than that, then unless you're incarcerated or disabled, you'll be assumed to be making that much. This can make it truly impossible to pay child support and have enough money to live. In this circumstances, there are very few options available to you. Child support arrears can never be discharged in bankruptcy, and they are still owed once the kids grow up and aren't eligible to receive support anymore. We've made a choice, as a state, to destroy people's lives if they can't pay child support.
If you fail to pay support for long enough, the state will revoke your drivers' license. If you willfully refuse to pay, they can even put you in jail (although this is very rare, and generally requires /willful/ disobedience).
Your best bet may be to talk to the Child Support Program administrator. The lowest amount of support anyone pays is $100 per month. Most people can usually afford that. If you can get your support adjusted to around this level, you can start making payments regularly, which is all they really need to show you're in compliance. But it can be very difficult to get this amount set if you already owe substantial arrears.