Not much in the way of legal trail w/in biking distance.
There is Ridgeline trail in south eugene if you like riding gravel hiking trails, and wild iris in SW eug. is open to bikes but is crowded with hikers so I don’t recommend it. Check the Eugene Parks map as many trails are hiking only.
Thurston Hills in east springfield has limited trails open now due to wet muddy trails but expect all trails to open in a month or so. Check with Willamalane.
As might be expected in a metro area with so little single track to offer, there are several non-sanctioned trails close in to the city, but I have not ridden them so can’t tell you much about them.