Does there exist a stochastic process $X_t$, $t \in [0,\infty)$, such that $X_t$ is distributed according to some distribution $f(x)$ that possesses finite variance and such that $X_t$ and $X_s$ are independent for all $s \neq t$?
Unlike other questions similar to this one I do not demand continuity in sampling paths.
If it exists what would a measure of the sampling paths look like? What would the paths look like?
My intuition is that such a processes probably doesn't exist based on my understanding of white noise. I.e. white noise in continuous time would have the above properties, but my understanding is that we must integrate over white noise to have something that makes sense and why we must write SDEs as
$$dx = \mu dt +\sigma dB$$
rather than
$$\dot{x} = \mu + \sigma \phi$$
where $\phi\sim dB/dt$ is the white noise as $\phi$ is not really a coherent concept.
Can anyone help me out here?
Yes, of course, as guaranteed by the Kolmogorov extension theorem. For instance, one can take $\lambda^{\mathbb R}$ where $\lambda$ is Lebesgue measure on $[0,1]$. But, as the Wikipedia article makes clear, it's not very useful
(Here $T$ would be your $[0,\infty)$.) This subject is immensely technical, and is discussed fully in chapter 7 of Bogachev's Measure Theory. I hesitate to summarize the crux of what goes wrong, but one thing is that your underlying sample space (something like $\mathbb R^{\mathbb R})$ is not separable; this meshes poorly with the countable-ness built into $\sigma$-algebras.