A system of two material objects exerting gravitational force on each other possesses

This question is a duplicate, but the very simple answer is yes, absolutely.

Every atom has a gravitational force with every other atom.

Yes, the Cavendish device can quite easily measure gravity between two everyday objects, as long as they weigh about a KG or so ... It's that simple.

For even smaller objects, it might be difficult to measure - but so what? (Any number of physical qualities are difficult to measure, when very small or very large.)

A simple way to think about it - consider the Earth. It obviously has gravity right? Now consider one small piece of the earth (just some random piece of rock inside the Earth). All those little pieces ... have to have gravity right? Or else the whole thing overall, would not have gravity. That thought experiment can help you see that "even your pencil has gravity".

What is the current scientific consensus on this?

100.0% of all scientists believe all atoms in the universe attract all other atoms gravitationally.

Do small objects (like my pencil) exert forces on other small objects (like my eraser),

Yes, absolutely.

(Note that -- very simply -- your pencil is attracting every tiny little rock that makes up the Earth. Consider the Earth as being trillions of pencil-sized little rocks. So, that's gravity.)

or is it only large celestial objects that can exert a gravitational force?

No, it very much applies to every single atom.

Gravitational force, also known as gravity, is the force of attraction that pulls two objects together. Every matter that has mass exerts a significant amount of gravitational pull on its neighboring objects. In simple words, gravity is the force of nature that drags a body towards the center of the Earth or towards any other physical object. Gravity majorly depends on the mass of the objects and the distance between them. It is one of the strongest forces in nature. Gravity was discovered in the year 1687 by Sir Issac Newton. With the discovery of gravitational force, a number of unknown concepts were also introduced by Newton including the three laws of motion.

Examples

1. Stability of the Objects

The objects present on the surface of the earth do not levitate or float in the air. This is because of the presence of the gravitational force present between the objects and the earth. The cup that is kept on the table does not hover in the air and stays in the same position until disrupted by an external force. Similarly, gravity is responsible to keep other objects in a stable position.

2. Tides

Tides are the short term periodic rise and fall of water in oceans. These large waves are caused in the oceans due to the gravitational pull of the moon and the sun exerted on the water present in oceans. In absence of the gravitational pull, the oceans would be calmer as the size of the tides would reduce to one-third of their original height.

3. Daily Life Activities

A number of daily life activities involve the application of gravitational force. For example, playing, sliding, jumping, running, etc. are completely dependent on gravity. A child slides through a slide only due to the presence of gravity on earth. If someone tries to perform the same sliding process in the space, which is a zero-gravity region it would not be possible.

4. Revolution of Celestial Bodies

The celestial bodies present in space revolve around the sun and rotate at their respective axis. The force which is accountable for the proper alignment and revolution of planets around the sun is the gravitational force. The same force is responsible for the revolution of the moons around their respective planets.

5. Fruits falling from trees

When the fruits growing on a tree get fully ripe, they tend to fall on the ground naturally. Interestingly, this process itself is responsible for the discovery of gravity. The idea of gravity hit Sir Issac Newton’s mind when he observed an apple falling from a tree. It was then that he pondered over the question that if the fruits fall from the tree, will the moon also fall. This iconic tale of Sir Issac Newton and the apple tree is very popular in the scientific world.

6. Falling Objects

Have you ever wondered why the objects falling on the ground do not fly in the air but fall straight on the ground? This is mainly because of the presence of the gravitational force. The mass of the object is directly proportional to the force of gravitational attraction. It means for heavy objects the gravitational pull will be stronger as compared to light objects.

7. Pouring Drinks

The drinks poured in a glass stay to the base and do not rise up to the brim. The gravity or the gravitational force makes this possible. Pouring liquids in space or in zero gravity regions is a tedious task as no force pulls the molecules to the base of the container.

8. Rolling Objects 

Rocks and other objects placed on a slanting surface tend to roll down if no obstacle is present in the path. This rolling of any physical body takes place because of the gravity in action. The objects kept in the same slanting position do not roll if they are present in a vacuum or in a zero-gravity region.

9. Feather eventually falling on the ground

Feather is a very light body that floats in the air. This light feather; however, eventually falls on the ground. The gravitational force is solely accountable for this to happen. A similar phenomenon can be seen with any other light body such as a paper piece.

10. Walking 

Walking is one of the most common physical phenomena that almost every living terrestrial animal follows. While walking our feet are in direct contact with the ground, we do not float or practice levitation. The ultimate phenomenon of gravity helps us to walk.

11. Gases in the Sun

Sun is the biggest star of our solar system. It is the central and most important heavenly body. It is a source of abundant energy. It is majorly a combination of gases and vapors. The gases present in the sun do not leave the celestial body because of gravity. In the absence of gravity, the gases will leave the sun’s surface, and we shall be left with a huge void in the entire system of planets and asteroids.

12. Precipitation

The water evaporating from the earth’s surface form huge clouds of water vapors. These water vapors then return to the earth by following a basic process of condensation. The precipitation is usually in the form of rainfall or snowfall. The rain or the snow falls directly on the ground and does not float in the air. Hence, the presence of gravitational force can be observed clearly.