$\mathrm{G}_2$ is the only one of the exceptional groups that can be defined as the stabilizer of a `generic' tensorial object on a vector space and, over the complex numbers, even this is not quite right.
More precisely, let $V$ be a vector space (over $\mathbb{F}$, which could be $\mathbb{R}$ or $\mathbb{C}$) of dimension $d$ and let $\mathrm{Sch}(V)$ be some Schur functor applied to $V$. For example, $\mathrm{Sch}(V)$ might be $S^k(V^*)$, the symmetric homogeneous polynomials of degree $k$, or $\Lambda^k(V)$, the exterior $k$-forms of degree $k$. Or it might be something less standard, such as the kernel of the natural map
$$
W:\Lambda^k(V)\otimes V \longrightarrow \Lambda^{k+1}(V)
$$
or the natural trace map
$$
\mathrm{tr}: V\otimes V^* \to \mathbb{F}.
$$
It is very rare that $\mathrm{GL}(V)$ acts on $\mathrm{Sch}(V)$ with open orbits outside of the classical cases, such as
$$
\mathrm{Sch}(V) = V,\ V^*,\ S^2(V),\ S^2(V^*), \Lambda^2(V), \Lambda^2(V^*)
$$
or these tensored with some power of the $1$-dimensional representations $\Lambda^d(V)$ or $\Lambda^d(V^*)$. When $d=2$, you also have $S^3(V)$ and $S^3(V^*)$, and when $d=6,7,8$, you have $\Lambda^3(V)$ and $\Lambda^3(V^*)$ (and, of course, these tensored with some power of the $1$-dimensional representations $\Lambda^d(V)$ or $\Lambda^d(V^*)$). That's about it.
When $\mathrm{GL}(V)$ acts on $\mathrm{Sch}(V)$ with an open orbit $\mathcal{O}\subset\mathrm{Sch}(V)$, we say that an element $w\in\mathcal{O}$ is stable. The $\mathrm{GL}(V)$-stabilizer of such a stable element $w$ is the group $G_w\subset\mathrm{GL}(V)$.
In the classical cases, these stabilizers are essentially the orthogonal and symplectic groups. In the case $d=7$ and $\mathrm{Sch}(V)=\Lambda^3(V^*)$ or $\Lambda^3(V)$ (or these twisted by some power of $\Lambda^7(V)$ or $\Lambda^7(V^*)$), the identity component of the stabilizer of a stable element is $\mathrm{G}_2$. These are the only cases in which the stabilizer of a stable element is (up to finite extension) an exceptional group.
The other exceptional groups $G\subset \mathrm{GL}(V)$ do occur as stabilizers of non-stable elements of some $\mathrm{Sch}(V)$. For example, when $d=26$, there is an element of $S^3(V^*)$ whose stabilizer has identity component $\mathrm{F}_4$, when $d=27$, there is an element of $S^3(V^*)$ whose stabilizer has identity component $\mathrm{E}_6$, when $d=56$ there is an element of $S^4(V^*)$ whose stabilizer has identity component $\mathrm{E}_7$, and when $d=248$ there is an element of $\Lambda^3(V^*)$ whose stabilizer has identity component $\mathrm{E}_8$. None of these are stable elements, though.
(These are the answers when $\mathrm{F}=\mathrm{C}$. The story in the case of real forms is more complicated. For that, you should look in Cartan's 1913 paper classifying the real forms of the complex simple groups.)
