Would space-based telescopes need to remain in the shadow of Earth to avoid light interference from the sun?
Its a nice idea, but it wouldn't work.

To stay in the Earth's shadow a telescope would need to orbit the Earth in the same time the earth orbited the Sun so that it always stayed on the opposite side. For its orbit to be this long it would need to be a long way out, remember geosynchronous orbit (one orbit in 23h 56min 4sec) is at a distance of 42,164 km. For a 1 year orbit the telescope would be so far away that its would not be in orbit around the Earth but the Sun, as it would have a much larger gravitational influence. It would also be too far away to be in the Earth's shadow cone.