If you want a physical barrier without blocking the view, a split rail fence might be a good way to go, and it it is the cheapest wooden fence option there is.

 If you want something solid to keep the pets in, a traditional picket fence might be a good option. This was built on site to match the terrain.

If you want a physical barrier to keep the pets contained, but don’t want to completely block the view, a 1×6 wood fence with green vinyl dipped mesh might be a good way to go.