Identify $\mathbb R$ as the set of functions $f : \mathbb N \to \{ 0,1\} $.
Then any sequence $\{ x_n \}$ becomes a sequence $\{f_n \}_n$ where $f_n : \mathbb N \to \{ 0,1\}$. But then, this is simply a function $g : \mathbb N \times \mathbb N \to \{ 0,1\}$:
$$g(m,n) =f_n(m) \,.$$
This way you can construct a bijection from the sequences of real numbers to the set of functions from $\mathbb N \times \mathbb N \to \{ 0,1\}$. Now, since $\mathbb N \times \mathbb N$ and $\mathbb N$ have the same cardinality, you get a bijection from the sequences of real numbers to the set of functions from $\mathbb{N} \times \mathbb{N} \to \{ 0,1\}$, which is just $\mathbb R$.
The reason that spaces of square integrable functions arose in the first place was to study the orthogonal trigonometeric (Fourier) series. Interestingly, Parseval had already noted in 1799 the equality that now bears his name:
$$
\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)^{2}\,dx = \frac{1}{2}a_{0}^{2}+\sum_{n=1}^{\infty}a_{n}^{2}+b_{n}^{2},
$$
where $a_{n}$, $b_{n}$ are the (Fourier) coefficients
$$
a_{n}=\int_{-\pi}^{\pi}f(x)\cos(nx)\,dx,\;\;\; b_{n}=\int_{-\pi}^{\pi}f(x)\sin(nx)\,dx.
$$
This comes out of the orthogonality conditions for the $\sin(nx)$, $\cos(nx)$ terms in the Fourier series. No definite connection was seen between Euclidean N-space and the above at that time; such a connection took decades to evolve. But square-integrable functions gained interest in the early 19th century, and especially after the early 19th century work of Fourier.
It took some time to see a general Cauchy-Schwarz inequality, and to begin to see a connection with geometry, eventually leading to inner-product space abstraction for the space of square-integrable functions. The CS inequality wasn't widely known until after the 1883 publication of Schwarz, even though essentially the same result was published in 1859 by another author. Hilbert proposed his $l^{2}$ space by the early 20th centry as an abstraction of the square-summable Fourier coefficient space, but also a abstraction of finite-dimensional Euclidean space. The connection with square-integrable functions was already firmly established.
In hindsight we can see good reasons that square-integrable functions are connected with energy, and other Physics concepts, but the abstraction seems to have been dictated more out of solving equations using 'orthogonality' conditions. Of course many of the equations arose out of solving physical problems; so it's also hard to separate the two. Now, after the fact, there is interpretation of the integral of the square of a function. On the other hand, the Mathematical abstraction of dealing with functions as points in a space, with distance and geometry on those points has been even more far-reaching, and a great part of the impetus for modern abstract and rigorous Mathematics.
Note: All of this happened before Quantum Mechanics.
Reference: J. Dieudonne, "History of Functional Analysis".
Best Answer
No, the cardinality of functions from $L^2(\mathbb R^n)$ to $\mathbb R$ is much bigger than the cardinality of $\mathbb R^n$ (for natural number $n$). So a bijection is not possible.
As a matter of definition, the set of functions from set $B$ to set $A$ is called $A^B$. Building on that we further define $|A|^{|B|} = |A^B|$. There is something to be proved about this definition, e.g. that it is well-defined regardless of the choice of representative sets $A,B$ for equivalence classes $|A|,|B|$ respectively, and that it agrees with our arithmetic notion in the case of finite cardinals. But for our immediate purposes we will assume that this is a sensible definition.
Your problem is then to compare $|\mathbb R|^{|L^2(\mathbb R^n)|}$ with $|\mathbb R^n| = |\mathbb R|^n = |\mathbb R|$. Written this way it appears that the former is much bigger than the latter since $L^2(\mathbb R^n)$ is an infinite family of equivalence classes of functions. But our intuition is sometimes not reliable about infinite cardinals, so let's exercise care in drawing that conclusion.
A useful step in comparing these cardinals is the proposition that if $2\le |A| \le |2^B|$ and $B$ infinite, then $|A^B|=|2^B|$.
To apply this to the first cardinal, we check whether $|\mathbb R| \le |2^{L^2(\mathbb R^n)}|$. Since $|\mathbb R| = |2^{\mathbb N}|$, and since:
$$ |\mathbb N| \le |L^2(\mathbb R^n)| $$
it is true that:
$$ |\mathbb R| = |2^{\mathbb N}| \le |2^{L^2(\mathbb R^n)}| $$
Therefore $|\mathbb R|^{|L^2(\mathbb R^n)|} = |2^{L^2(\mathbb R^n)}|$.
Finally since $|\mathbb R| \le |L^2(\mathbb R^n)|$, we can use Cantor's theorem to conclude:
$$ |\mathbb R| \lt |2^{\mathbb R}| \le |2^{L^2(\mathbb R^n)}| = |\mathbb R|^{|L^2(\mathbb R^n)|} $$
Your desire to have a bijection between $\mathbb R^n$ and the space of functions from $L^2(\mathbb R^n)$ to $\mathbb R$ might be met by some approximation scheme. This is a problem of analysis and approximation theory, and the right approach depends on the topology needed for your intended application.