Short answer: Every spherical variety is a Mori Dream Space.

Longer answer: Every projective variety is a "GIT quotient", but the most obvious construction is probably unenlightening. Let $X$ be a projective variety embedded in $\mathbb{P}^n$. Let $V$ be the affine cone over $X$ with the natural action of $G=\mathbb{G}_m$. Then $V$ is a $G$-invariant, Zariski closed subset of the affine space $\mathbb{A}^{n+1}$ that has the "standard" linear representation of $G$ (by scaling), and the GIT quotient $V//G$ is $X$.

However, the construction above depends on the choice of a projective embedding, or at least of an ample invertible sheaf, so it is not canonical. There is a class of projective varieties that are canonically GIT quotients, namely the Mori Dream Spaces first studied by Hu and Keel. Every flag variety, and indeed every projective variety homogeneous under a linear algebraic group, is a Mori Dream Space. In fact, there is a class of varieties that contains both projective homogeneous varieties and toric varieties (another large class of Mori Dream Spaces), namely "spherical varieties". Every spherical variety is a Mori Dream Space.

Short answer: Every spherical variety is a Mori Dream Space.

Longer answer: Every projective variety is a "GIT quotient", but the most obvious construction is probably unenlightening. Let $X$ be a projective variety embedded in $\mathbb{P}^n$. Let $V$ be the affine cone over $X$ with the natural action of $G=\mathbb{G}_m$. Then $V$ is a $G$-invariant, Zariski closed subset of the affine space $\mathbb{A}^{n+1}$ that has the "standard" linear representation of $G$ (by scaling), and the GIT quotient $V//G$ is $X$.

However, the construction above depends on the choice of a projective embedding, or at least of an ample invertible sheaf, so it is not canonical. There is a class of projective varieties that are canonically GIT quotients, namely the Mori Dream Spaces first studied by Hu and Keel. Every flag variety, and indeed every projective variety homogeneous under a linear algebraic group, is a Mori Dream Space. In fact, there is a class of varieties that contains both projective homogeneous varieties and toric varieties (another large class of Mori Dream Spaces), namely "spherical varieties". Every spherical variety is a Mori Dream Space.

Edit. The OP clarified that he wants $V$ to be an affine space, not an affine variety. For most Mori Dream Spaces, $V$ is not an affine space (for toric varieties, $V$ is an affine space, as follows from the theory of the Cox ring).

Every spherical variety is a Mori Dream Space.

Short answer: Every spherical variety is a Mori Dream Space.

Longer answer: Every projective variety is a "GIT quotient", but the most obvious construction is probably unenlightening. Let $X$ be a projective variety embedded in $\mathbb{P}^n$. Let $V$ be the affine cone over $X$ with the natural action of $G=\mathbb{G}_m$. Then $V$ is a $G$-invariant, Zariski closed subset of the affine space $\mathbb{A}^{n+1}$ that has the "standard" linear representation of $G$ (by scaling), and the GIT quotient $V//G$ is $X$.

However, the construction above depends on the choice of a projective embedding, or at least of an ample invertible sheaf, so it is not canonical. There is a class of projective varieties that are canonically GIT quotients, namely the Mori Dream Spaces first studied by Hu and Keel. Every flag variety, and indeed every projective variety homogeneous under a linear algebraic group, is a Mori Dream Space. In fact, there is a class of varieties that contains both projective homogeneous varieties and toric varieties (another large class of Mori Dream Spaces), namely "spherical varieties". Every spherical variety is a Mori Dream Space.