Not only could it be done, but you could in fact get the hardware right now.
Behold, the RB2000 rocket belt:
We can work out the probable $\Delta_v$ of the rocket via the good old rocket equation:
$$\Delta_v = g_0 \cdot I_{sp} \cdot \ln \left( \frac{M}{M_e} \right)$$
Where $g_0$ is a standard gravity, $M$ is the fully fuelled mass of the rocket and $M_e$ is its dry mass. High-test peroxide monopropellant has an Isp of ~140 seconds. Given a desired $\Delta_v$ of 153m/s to escape Mimas, you'd need a mass ratio of ~1.12.
If the rocket belt weighs 60kg (wikipedia says "equipped" so lets assume that's the wet mass) and it has 23l of peroxide fuel, the dry mass of the rocket alone would be 26.65kg. An average human is ~62kg. A fancy lightweight spacesuit might be 40kg (this is perhaps a little optimistic). This would give a fully dressed and fuelled up weight of 162kg, and a dry mass of 128.65kg. That gives you a mass ratio of ~1.26 and a $\Delta_v$ of 316m/s... more than enough (which is good, because I'm ignoring complications like gravity drag as being too much like hard work).
Obviously the RB2000 as it stands wouldn't work in a vacuum, but clearly it has enough thrust to do the job (it works on Earth) and more than enough fuel, and there's still plenty of scope for adding additional fuel tanks if needs be.
For the second part of your question, you can work out the size of a spherical body with a given escape velocity $v_e$ and density $\rho$ by fiddling with the simple escape velocity equation like this:
$$\frac{M}{r} = \frac{v_e^2}{2G}$$
$$M = \frac{4}{3}\pi r^3 \rho$$
$$r^2 = \frac{3v_e^2}{8\pi G \rho}$$
Where $v_e$ is escape velocity. With density similar to Mimas (1.15g/cm3) and an escape velocity of the rocket belt's $\Delta_v$ of 316m/s, you'll get a radius of about 394km, which is coincidentally about twice that of Mimas, and a mass of about 2.95x1020kg.
Obviously on such a small world the surface gravity will be lower (its about 0.013g) so you'd be able to carry a much heavier rocket which in turn would help you lift off from a larger body, but its hard to pin down precisely what counts as "a rocket pack small enough to carry on your person", as Hohmannfan's comment showed. A limit purely by weight $W_l$ gives rise to this inequality in terms of radius $r$ and density $\rho$:
$$\sqrt{\frac{8\pi G \rho r^2}{3}} < g_0 \cdot I_{sp} \cdot \ln \left( \frac{3W_l}{4\pi G \rho rM_e} \right)$$
which I shan't attempt to simplify. A weight limit of 1000N (and neglecting any increase in fuel tank mass, which is obviously lazy and wrong) will give you enough delta-V (2440m/s) to escape from Europa (surface escape velocity ~2025m/s) and a mass ratio of ~5.77, which is about ~423l of fuel... dubiously "small enough", but should give you a rough upper limit. The Moon is just a bit too hefty to escape from with this weight limit, but all smaller bodies with no atmosphere would seem fair game.
