I know that objects attract other object. The more mass the object, the more it attracts.
But what constitutes an object in this sense? For example the earth attracts me. But I can bend down and pick up a piece of earth. Of course the earth I picked up doesn't attract me as much as the earth beneath my feet, but it now attracts the earth in my hands with the same force as me.
All atoms or massive particles create a gravitational field around themselves with which each particle interacts. The Earth is acting as a single object (illusionary) because each particle inside of it is attracting every other particle inside of it. This is squeezing the earth together. If you pick up a hand of dirt, nothing really changes, but the position of a neglectible part of Earth's mass. That part of dirt is now attracting you sideways instead of downwards. But because of the huge difference of mass in your hand compared to that in the earth, the side pull is negligable considering the downpull of every other particle of dirt in the earth.
An object, as your question puts it, is anything with mass.
Think of it like a suitcase full of clothes. You can pull out a sock, or a thread off of a sock, and individually it doesn't have much mass.
Collectively, though, you've got a heavy suitcase.
I think what you are asking is when the gravitational force of attraction becomes significant so that it can be "felt".
That is entirely dependent on what other forces are present and their magnitude and direction.
You might not think that the gravitational attraction between a 150 kg lead ball and a 1 kg lead sphere is very large (~$10^{-7} \rm N$) but Cavendish arranged his apparatus in such a way as to make that very small force of gravitational attraction have a measurable effect on his apparatus.
Always competing with gravitational attraction is the electrostatic force which for two electrons is $10^{36}$ times larger.
The significant difference between the two forces in that the electrostatic force can be both attractive and repulsive means which means that for large bodies, which have approximately the same number of positive and negative charges, the gravitational force wins out over the electrostatic force although of course you do feel the electrons on the surfaces of the Earth repelling the electrons in the soles of your shoes when you are standing on the Earth.
For a space station it is impractical to make the space station so massive that it has an attractive gravitational force to allow astronauts to stop feeling weightless and so the better idea is to have the space station rotating as described here.
When you pick up a piece of the Earth the attraction felt by you, via the sensation of the piece of earth push down on your hand, is due to the force of gravitational attraction on the piece of earth due to the Earth not the very, very much smaller force of gravitational attraction on the piece of earth due yo your body because the mass of the Earth is so much greater than your body mass.
