For $(x, y) \in \mathbb{R}^2$ to be a constructible point, we must necessarily have $[\mathbb{Q}(x, y):\mathbb{Q}] = 2^r.$
According to a book I'm reading on Galois theory, questions about sufficiency are much more difficult. There is a nice exercise in the book to show that $\mathbb{Q} = F_0 \subseteq F_1 \subseteq \cdots \subseteq F_r = \mathbb{Q}(x, y)$ with $[F_{j + 1} : F_j] = 2$ is a sufficient condition for $(x, y) \in \mathbb{R}^2$ to be a constructible point.
But is just $[\mathbb{Q}(x, y):\mathbb{Q}] = 2^r$ a sufficient condition for $(x, y) \in \mathbb{R}^2$ to be a constructible point?