Is 338 breeding couples (all atleast more distally related to each other then first cousins) enough to establish a viable human population without genetic problems popping up in the near future?

I never did anything about it due to laziness, but about 6 months ago I had an idea for a sci-fi story that involved aliens abducting 676 people (age 16 to 27) from the UK and dropping them off on another planet. Assuming they had monogamous marriages would that be enough to establish a civilisation?

Almost certainly yes. Depending on the circumstances in time genetic drift will occur and providing separate villages/towns are setup with marriage between towns rather within a town then all should be well.