Is there any orbit at which the Roche limit can be "felt"? Do any of the planets have a Roche limit that is strong enough to be felt by an astronaut whilst in orbit?
Roche limit happens where the gravity of the object, trying to pull the object together, becomes smaller than the tidal force (trying to pull the object apart).
But the astronaut is bound by not gravity, rather by the electromagnetic interaction between his/her atoms. The own gravity of the astronaut is negligible, compared to the electromagnetic interaction.
However, the tidal force affecting an astronaut, should require a little calculation. We can derive the formula of the gravitational acceleration around a point-like body ($F=\frac{GM}{r^2}$), we get
$$\frac{dF}{dr}=\frac{2GM}{r^3}$$
(We can ignore the sign on obvious reasons.)
Here $G$ is the gravitational constant, $M$ is the mass of the body, and $r$ is the distance.
Substituting the values of the Sun, we get $\frac{2\cdot 6.67 \cdot 10^{-11} \cdot 2 \cdot 10^{30}}{(7\cdot 10^8)^3} \approx 7.78\cdot 10^{-7} \frac{\mathrm{m/s^2}}{\mathrm{m}} \approx \underline{\underline{8 \cdot 10^{-8} \frac{g}{\mathrm{m}}}}$.
More clearly, if we are orbiting the Sun just above its surface, a roughly 2m long astronaut feel that his head and foot are pulled apart by around $1.6\cdot 10^{-7}g$ weight. In the case of a $70\:\mathrm{kg}$ astronaut, it is around the weight of $0.0112$ gram on the Earth.
The astronaut wouldn't feel it, but not very sensitive sensors could already measure it.

This calculation sometimes used $\mathrm{g}$ for &quot;gram&quot;, as unit of mass, and $g$ as the (non-standard) unit of acceleration.