Work with a national real estate company. They can find you rental homes or apartments. Tell them your specific requests (pets allowed etc).
If you have friends in area, ask them about places for rent.


Monster.com for submiting your resume.
Contact the state unemployement office in Eugene, they post jobs.
Would your present employers transfer you to Eugene? For example, if you worked for a national retail chain, transfer to that city.

These few tid bits are off top of my head at moment.