So gravity turns things round It makes sense, since gravity tends to push the surface of a body towards it's center. Unless I'm mistaken, everything with mass has it's own gravity, every atom and for instance, our own bodies should also have their own gravity. The question is: how strong is our own gravitational pull? I know it must be extremely weak, but is there actually anything at all that gets attracted to us, like maybe, bacteria or molecules?
And finally (this will sound ridiculous, but I'd really want to get an answer or at least a way of calculating it myself): What size would a human body have to reach in order for it to collapse into a sphere?
 A: 
It makes sense, since gravity tends to push the surface of a body towards it's center

Yes, gravity tends to pull towards the center of mass. I think you're comparing human body to celestial bodies - for instance, stars. Stars collapse within themselves after their lifetime because, their internal (thermal) pressure is not sufficient enough to sustain the gravitational collapse.

How strong is our own gravitational pull? I know it must be extremely weak, but is there actually anything at all that gets attracted to us, like, maybe, bacteria or molecules?

Yes, it's extremely weak. The constant $G$ is already making the situation very weak. When you mention bacteria or molecule, you are actually reducing the effect further due to their mass. Gravity doesn't depend upon the size. Only MASS matters here. As nitrogen is around us mostly, you can calculate the force between you and a single nitrogen molecule at a distance of an angstrom, which is barely about $F\simeq 10^{-15}\text N$.

What size would a human body have to reach in order for him to collapse into a sphere?

Wiki has a nice quote on gravitational collapse:

Because gravity is comparatively weak compared to other fundamental forces, gravitational collapse is usually associated with very massive bodies or collections of bodies, such as stars (including collapsed stars such as supernovae, neutron stars and black holes) and massive collections of stars such as globular clusters and galaxies.

Human body is very small compared to any celestial object (just like the size of an atom in a large city). So, you'd probably need the size of some bigger (massive) celestial object to have a significant effect on the gravitational collapse.
A: Well, more or less, the size of a planet... :) There are fu*** big asteroids out there that aren't big enough to get spherical shape. Also, it will help if most of the planet is gas or liquid.
You can calculate the forces classically by Newton's expression for gravitational force: $$\vec F=-G\frac{Mm}{r^2}\vec u_r$$
$G$ is Newton's constant: $G\approx6.67\cdot 10 ^{-11}$ in IS units. You can see it's really week, compared to the constant of electrical force: $k\approx 9\cdot10^9$.
$M  $ and $m$ are the two masses between you want to calculate the force. $r$ is the distance between them. $-\vec u_r$ indicates the force is attractive.
So, the size of the person, well, a million things happens before the size is big enough to collapse in a sphere, like broken bones. You would need to know how resistent tissues are before they brake and so no, but I would go for a good approximation, that it's the order of the size of the moon (probably less than that), like 2 or 3 orders og magnitude smaller.
