Yup.
I've been riding bikes since I was 4. But I never rode on the streets with traffic (not counting just cruising around my own neighborhoods with little to no traffic) until I was 39, a full year after I started bike commuting to work where I purposely took side roads and back paths to avoid streets.
At first it was absolutely horrifying. By the 3rd or 4th commute once I started riding in the streets I was over it.
It should always be a LITTLE scary. You don't want to be terrified or you'll be too tense and likely to do something stupid. But you want to be a LITTLE nervous because it'll keep your guard up and keep you focused. If you get TOO relaxed when riding in the street you're likely to get distracted and miss approaching danger. You should be just nervous enough to keep you paying attention, but not enough that you can't enjoy the ride.
Anyway...point is...I was 39 when I started riding in the streets. I got over the near paralyzing fear of a car passing me pretty quick. So yes, it can be done at 35.