Why doesn't Einstein's general theory of relativity seem to work on Earth? I am new to physics and I have learned a little bit about gravity from Einstein's perspective. The gist is that heavy objects create curvature of spacetime, and free-falling objects move on the straight lines in the curvature. 
But I am failing to understand how this applies to objects on Earth. For example, why don't elephants make spacetime curvature and cause dust to go around them? Or simply, how does spacetime curvature work inside a planet?
 A: The short answer: general relativistic effects are mostly not noticeable on such a small scale (except in a few cases). For example, one common way to characterize the strength of a gravitational field is through the dimensionless number:
$$A=\frac{GM}{Rc^2}$$ 
where $G = 6.67 \times 10^{-11} \text{ }\mathrm{m^3 kg^{-1} s^{-2}}$ is the gravitational constant , $M$ is the mass, $R$ is the distance to the object, and $c=2.99\times 10^8\text{ m/s}$ is the speed of light. For being directly next to an African Bush Elephant:
$$A \approx 10^{-24}$$
For the Earth: $A \approx 10^{-9}$
and even for the Sun, $A \approx 10^{-6}$. Generally, when $A \ll 1$ the effects are practically negligible. In fact, most of the physics that describes planetary motion in our solar system can be accurately described with standard Newtonian physics. However, there are some extremely sensitive cases (e.g. orbit of Mercury / satellite communication) where we need to take into account general relativity.
Just for comparison, a black hole (using the Schwarzschild radius) gives a value of $A=0.5$, which is much stronger than our Sun's meaning we definitely will need to take into account general relativity.
A: Instead of spacetime curvature, it is better to think of a different system of coordinates.
For example, if someone is at 1 km from the North Pole, all the land (or ice) until the horizon in all directions (a radius about 5 km) can be approximately by a flat surface.
Even being flat, during any path in straight line, except if it is radial to the pole, the route is not constant, if by route we understand a given direction ($221^\circ$ for example, when $0^\circ$ is direction North. It is a consequence of using polar coordinates. But when the calculations are corrected for the effect of that curvilinear coordinates (covariant derivative) they show a constant straight velocity.
In a similar way, in our daily experience of a constant $g$ (approximately flat spacetime), any falling objects has a non uniform velocity in our coordinates of space and time (and also according to our senses in this case). But it also follows a straight constant velocity, in the meaning that the covariant derivative of the velocity is zero. 
A: Of course it does work on earth. Just consider these facts:


*

*General Relativity also describes/encompasses 'standard' gravity. Just in a slightly over-complicated way.

*Things 'thrown in the air' as well as satellites follow exactly the
'straight' lines (aka geodesics) of general relativity.

*The GPS system needs to account for general relativistic effects in order to achieve its precision.

*There have been experiments of transporting clocks around in planes that exactly confirmed general relativity.


Sure, it is not as dramatic as displayed in documentaries, but often in science the spectacular is hidden in the details.
A: 
Or simply, how does spacetime curvature work inside a planet?

The curvature of spacetime is very much observable on Earth. Indeed, were spacetime not curved, then undisturbed objects would follow straight lines in spacetime, in other words, they would move with constant velocities; but objects released above the surface of Earth very visibly deviate from that, exhibiting acceleration towards the planet.
As for phenomena on Earth that can be explained by GR but not by Newtonian gravity, we can now make such tremendously accurate clocks that we can use gravitational time dilation to use them as glorified elevation measuring devices.
A: Elephants do have stress-energy and do create spacetime curvature.
Here on Earth on the small scales, the other forces (EM, weak, strong) dominate over gravity.
Still, you could ask, what would happen if you put an elephant into space (zero gravity) and put dust around it. Will the dust be gravitationally attracted to the elephant? Yes it will. This is how originally celestial bodies started to form from dust. 

In astrophysics, accretion is the accumulation of particles into a massive object by gravitationally attracting more matter, typically gaseous matter, in an accretion disk.[1][2] Most astronomical objects, such as galaxies, stars, and planets, are formed by accretion processes.

https://en.wikipedia.org/wiki/Accretion_(astrophysics)
Now why do we not see the same effect here on Earth of an elephant having visible gravitational attraction on dust around it? It is because here on Earth, the spacetime curvature is dominated by the Earth's gravitational field, and it points towards the center of Earth. Dust moves (disregarding atmosphere) towards the center of Earth.
A: It works pretty much. We just happen to live in an environment where Newtonian approximation of the GR is good enough.
We use the Newtonian approximation


*

*because of the tradition,

*because it is much, much less of a math hassle, and

*because the results of the approximation are of acceptable accuracy


As for why elephants don't cause object to orbit around them: 
first, see https://en.wikipedia.org/wiki/Hill_sphere .
For short: the elephants in the general case are in the Earth's Hill sphere and it is the Earth here that distorts the spacetime the most. If you want an elephant-sized object to have its own satelites, you have to put it somewhere far away from Earth, so its Hill sphere to be larger than the object itself.
second, the air. Air is interacting with everything that tries to orbit the Earth in rather strong manner. Because of the air, every Keplerian orbit becomes a ballistic curve and the object goes to the Earth surface where the interaction is even stronger than in the air.
A: The effects at the human scale is normally too small to be measured but it does happen.
Time dilation has been measured by using two accurate clocks one installed at ground level and one installed on a tower (see: Hafele–Keating_experiment). So that is one way general relativity affects real things at near-human scale on earth. Based on the experiment we can be confident that people at the top of skyscrapers experience time differently than people on the street.
More recently researchers have managed to use even more accurate clocks to measure time dilation by moving the clock a few centimeters vertically. (Also see: https://arstechnica.com/science/2020/04/portable-clock-provides-new-ruler-for-measuring-the-earth/)
