Why don't you feel gravity the same way you feel a car's acceleration? If you are in an accelerating car or spaceship you feel an apparent force pushing you backwards, which as I understand is due to your own body's inertia (it wants to keep its current velocity but the vehicle's velocity keeps changing as it accelerates). You don't feel this backwards/upwards force with gravity.
Is it because gravity acts on your whole body equally? In contrast, if you are in a car technically only the part in contact with the seat would be accelerating at the same rate as the car (and the rest of your body will be pushed back due to wanting to keep its own inertia)? While with gravity every point of your body has the same inertia so you don't feel an opposite push?
If you were enormously tall, would your head keep feeling this backward force as the gravity acting on your body would no longer be uniform?
 A: You DO feel the acceleration of gravity.  For instance, as I sit here typing this, I feel my butt pressing into the seat of my chair with a force of 1 g.  If I stand up, I will feel my feet pressing on the floor with the same force; likewise if I go do a yoga headstand, I will feel that 1 g acting in the opposite direction.  However, you may not NOTICE the force unless you choose to pay attention to it.
The reason you notice accelerations in a car more is that your brain tends to react more to changes in stimuli rather than to steady values.  If you're driving on a winding road, the acceleration you experience changes from moment to moment, as the acceleration vector shifts from side to side.  Make the same sort of turns if the road is banked just right, or in a light plane where you can bank for yourself, and you don't experience those side-to-side accelerations, only a much less noticable increase in the force acting vertically with respect to your body. That is, you just sit a bit heavier.
A: 
You don't feel this backwards/upwards force with gravity.

You have this a little confused. In an accelerating car from the car’s non-inertial frame there is a contact force pushing forward and an inertial force pushing backward. In the ground’s frame there is a contact force pushing upward and a gravitational force pulling downward.
So the analogy is between the gravitational force and the inertial force. Thus you should say “you don’t feel this backwards/downwards force with gravity”. The backward inertial force for the car is analogous to the downward force of gravity, not the upward contact force.
In both cases the feeling is analogous. The easiest way to see that is to examine accelerometers. An accelerometer in the car detects the forward acceleration from the contact force only and not the backward inertial force. Similarly, an accelerometer on the ground detects the upward acceleration from the contact force only and not the downward gravitational force. Inertial/gravitational forces are not “felt”. What you feel in both cases is only the contact forces.
A: They are both exactly the same and feel exactly the same.
In fact, they don't feel like anything. Gravitational force and centrifugal "force" or other inertial "forces" cannot be "felt" at all. Because (in the reference frames where they appear) they act on every single particle in your body at the same time.
When you jump from a plane you are speeded up by the gravitational force. Each particle in your body is. The only reason you don't feel it is because all the particles of your body move equally. But hit a bird with your arm on the way down, and suddenly your arm is slowed down and not keeping the same speed as the rest of your body. You feel that as your tissue molecules being stretched and moved relative to each other.
Feeling and pain in your body comes not from force acting on your body parts but from relative motion between body particles.
That's the key. It is never force, you feel. It is always only the effect of the force you can feel. It is the acceleration or relative velocity between body parts and particles you feel. Only. So the way we feel and experience forces is from how they cause changes in the otherwise equal motion of particles within us.

*

*You can certainly feel gravity when you hit the ground. The first-impacting particles are slowed down suddenly while the rest of the particles are still moving downwards.


*Likewise, you can feel the centrifugal effect only when squeezed towards the car door. Or when your seat pulls in you. Your body tends to keep moving straight due to its inertia, so when the car turns then it is not you who is being forced or pushed or pulled - it is the car that is moving. If you jumped off from the seat, and then the car turned, you wouldn't feel a thing.
Inside the turning car, it looks like you moving towards the car door because your brain tricks you into assuming your surroundings to be stationary. But in reality ti is the car door which is moving towards you. It is an illusion. You are not being influenced at all - not until the car door hits you, that is, and only then do you feel anything.

Sidenote: As far as I'm aware, the very fact that gravity was observed to "feel" and behave just like any fictitous pseudo-force such as centrifugal forces and Coriolis forces etc. lead to a belief that perhaps gravity is a fictitious force as well. Perhaps we are living within a non-inertial reference frame. This has turned out to be explained via gravitational fields in general relativity.
So, suddenly, just like the centrifugal "force" is nothing but an illusion it seems that gravity is nothing but an illusion too. (They may be illusions, but their effects are of course clearly still very, very real - what I mean by illusion is that they are inertial tendencies rather than actual forces; we are just tricked to think of them as forces.)
A: You do feel gravity. I dont know if you ever been in a zero gravity situation (skydiving, a rollercoaster, jumped to the water off a big jump etc), but those situations are exactly what you should feel if you were in outer space. Indeed, astronauts in ISS are constantly in a free fall. So the answer is that you do feel gravity, its just that you are very used to it.
I don't know what you would feel of you were that tall, it's difficult to imagine! But you may feel tidal forces along your body depending on how tall you are and how you are oriented with respect to the body.

Edited for your comment:
@Veirian, indeed that's correct, you are not missing anything. You cant distinguish the gravitational pull from earth from that of an accelerating body (assuming no tidal forces are around: i.e. in a sufficiently small volume). When you are standing on Earth, you feel a force of 1g, you might not be moving but you are surely accelerating for a non accelerated motion (free fall) would be an elliptical orbit around Earth. But you are standing still! Hence you are not on free fall and thus are experiencing an acceleration away from Earth, identical to that of a rocket at 1g.
A: The answer to your question is more psychology than physics.
There is never a moment that you are not subject to gravity so your adjustment to gravity is completely automatic, completely internalized.
When you are standing up your feet are compressed by the weight of your entire body. Your neck is compressed only by the weight of your head.
Conversely, when you balance yourself upside down then your neck has to support the weight of your entire body.
The difference is huge, but your feet do not complain about being compressed in the same way that your neck will complain. You kind of  know your feet are compressed, but you don't pay any attention to that.

To create 1 G of acceleration in horizontal direction you need some vehicle capable of 1 G of acceleration. Today's high performance cars are capable of that. If the car can go from 0 to 100 kph (60 mph) in less than 2.7 seconds then the car is pulling more than 1 G of horizontal acceleration.
Take a model that offers the option of folding down the rear seats to create sufficient space to lie down. Lie down on a roller so you are free to roll towards the front or the rear.
Then when the car changes velocity in one direction 1 G worth of accelerative force is applied to your feet, and in the other direction 1 G worth of accelerative force is applied to the top of your head.
A: 
You don't feel this backwards/upwards force with gravity.

To begin with, can you feel forces at all? What is it that you feel? You can feel pressure if something tries to deform your body slightly (or not very slightly), and this is typically caused by an outside force acting on you.
If the car is accelerating, it, including the back of the seat you’re occupying, is trying to increase its velocity while you’re trying to maintain your current velocity. So the back of the seat is pushing you, and you feel this pressure.
If the car is anywhere close to a massive body such as a planet, it’s trying to get closer to that body. If it’s on the surface of the planet, the surface is trying not to get deformed too much and not to let the car fall through it so it exerts upwards pressure on the car, which it, in turn, exerts on your buttocks.
So you feel the acceleration and the gravity in exactly the same way, i. e. you don’t feel them directly; and you feel their effects in exactly the same way, i. e. as pressure from the seat.
See other answers for how it would feel if the car is in fact not on the surface of a planet.
A: You are looking at balanced forces vs unbalanced forces. Gravity can be considere somewhat constant force that you counteract with your muscles, if I consider a person who is standing up they will feel a net force of zero, gravity is completely balanced by their leg and glutes muscles. When the car accelerates it generates a force via Newton 2nd principle $F = ma $. This force is literally the thing that pushes you forward. At this point you are no longer in an inertial reference frame (the sum of all forces is zero) so you will feel a backwards facing force, balanced only by your contact with the car's seat (which is generally speaking deformable).
If you were enormously tall your head would not feel any backwards force as gravity will always point downwards. That is to say that if gravity would not be constant on your body you would have a non null stress tensor that would be accounted to have stability and not collapse.
A: 
You don't feel this backwards/upwards force with gravity.

Ever been in a rollercoaster? That stomach feeling you get after your crest a hill at full speed and start dropping? That's gravity accelerating you.
It's the exact same feeling you get when being in an accelerating car. Just a different direction and intensity.

What I meant was that in a spaceship for example that accelerates upwards you feel a force pulling you downwards, but on Earth even though gravity acts downwards you feel a force pulling you down, rather than one pushing you up.

You've misunderstood the purpose of the relativity example. The takeaway isn't that you feel an opposite force to what you should. The takeaway is that whether A pushes against B, or B pushes against A, both scenarios feel the same to A and B.
Therefore, to apply this analogy to Earth, you'd have to consider two scenarios, one where Earth is static and has gravity (= you push against the Earth) and one where there is no gravity but the Earth is accelerating in the direction you are (= Earth is pushing against you).
There is no "it feels like it's the other direction" sensation in the relativity example that would relate to your "gravity should feel like it's upwards" claim. I suspect you've glossed over the fact that in the different scenarios, different objects (i.e. either A or B, depending on the scenario) are the ones applying a force (in opposite directions).

Is it because gravity acts on your whole body equally?

The only difference here is that gravity does not contort you or imbalance you, where an accelerating car might (e.g. your head tilting back). However, for any reasonable car ride, there is no difference between the car accelerating you and gravity accelerating you.
I suspect you're misunderstanding the "acceleration" part. When you're standing on the ground, you're not accelerating. When the car start going faster and faster (and so do you), that is acceleration. These are two different sensations because one is acceleration and the other is not.
But if I drop you from a great height, and you start falling downwards, that is the same feeling as if you're in a car accelerating at 9.8 m/s², because now both scenarios are a form of acceleration.
You cannot equate (and therefore directly compare) gravity's pull (while standing still on the ground) and the acceleration gravity's pull might cause (e.g. when not touching the ground). These are two different things, and you experience them differently.

If you were enormously tall, would your head keep feeling this backward force as the gravity acting on your body would no longer be uniform?

The size you'd have to be for any such effect to become noticeable, you're already running into the issue that your head must move more than your feet, because you're moving around the Earth's surface while trying to remain level to it, which means that you're rotating your position as you move along the surface, and your head has to travel much more distance than your feet, so you're going to feel like you're off-balance.
This effect will completely overshadow a tiny difference in gravitational pull (you'd be surprised just how far out you have to go for it to meaningfully lower).
A: Gravity can be described as a pure acceleration, and it is perceived at such. You'd feel 0G when in free fall and when flying at constant velocity in outer space, and you'd feel 1G when standing still on the ground. If, in addition, you accelerate forward in a car with 1G, your total acceleration would amount to ca. 1.4G.
However, whereas your body senses all sources of acceleration in the same way, it is not as well accustomed to non-gravitational accelerations, its non-zero derivatives and certain related sensor inputs, as it is to gravity. The change of feeling moreover intensifies when sensor signals do not meet with your expectations or each other, or when they have high derivatives.
As for your premise that the difference in perception of acceleration is caused by the fact that gravity applies to your body as a whole whereas the  acceleration in a car is the result of a contact force, it is not correct. The force balances are similar. In both cases the contact force is in equilibrium with an inertial force as the result of an acceleration of every point in your body. With gravity, the contact surface (when standing) is the bottom of your feet, on which a force is exerted equal to that required for accelerating every point in your body up with 1G. When accelerating in a car, the contact surface is mainly your back, on which a force is exerted equal to that required for accelerating every point in your body forward with the actual acceleration. The problem in your thinking might be related to a swap of cause and effect (gravity causes the reaction force; car acceleration is caused by the reaction force), even though Newtonian physics don't actually care about this. After all, force balances are about instantaneous equilibria: every action is also a reaction, and vice versa.
I'll break the main answer down in several points for the interested reader.
The human body does in fact sense acceleration and thus gravity directly, in contrast to what most answers here claim. 
I am surprised to read all the claims stating that humans cannot sense acceleration directly. This is incorrect. The vestibular system, located behind your ears, allow you to directly sense accelerations. More specifically, it consists of the otolith and the semicircular cannels, which allow you to sense linear and rotational accelerations respectively. They help you balance, and they are responsible for making you feel nauseous after jerky rollercoaster rides. If you want to know how it feels like to have a malfunctioning vestibular system: get drunk*. And no, the vestibular system does not distinguish gravity from acceleration, for the fact that gravity is an acceleration. By combining the gravitational acceleration and that of a car, the amplitude of the acceleration vector increases and its orientation changes.
For your interest: 
There is another important sense in the human body of which many are not aware, namely proprioception. It senses internal body kinematics and kinetics, using the golgi tendon organs in your muscles. They close the loop with muscle control. The role of proprioception in daily activities such as walking is very important and shouldn't be confused with touch. Watch this short 2 minute video to understand how it's like to go around without proprioception.
The human body is a master of sensor fusion, but it can be fooled. 
Your central nervous system fuses many types of sensor signals and your anticipation/expectation/knowledge for purposes of understanding and controlling all sorts of activities. These sensor signals include touch, vision and the aforementioned acceleration and proprioception. When you hit the accelerator in your car, you'll feel relatively comfortable. Everything matches up: the acceleration, the compression in your seat, the vision and your own anticipation. Instead, when you are a co-driver, it will easier feel less comfortable, because you lack the anticipation.
For your interest: 
There are also situations where sensor signals do not match, which cause confusion and discomfort. Consider for example being in a parked train, and the train(s) next to you start driving. Your vision might make you think that you start moving, but your vestibular, touch and proprioceptive senses disagree, causing you to feel discomfort up to the point that you realise that your sensor fusion made a mistake. These illusions of self-motion are well known and called vection. The initial mistake occurs because your sense of vision is typically given more value. That said, your sensor fusion system can actually filter a non-matching sensor signal quite well. This has allowed for inventions such as flight simulators to feel realistic. The vision, sound, surrounding (realistic cockpit) and seat compression as caused by slightly tilting the platform all make you think that you are flying a plane. The only missing aspect is the actual linear acceleration (although not entirely, because the platform can move back and forth several meters, and mimic vibrations such as those caused by turbulence).
Over the long haul, the human body is a master of adaptation.
So why is it then, that we feel the effects of accelerating in a car more prominently (even if we hit the accelerator ourselves) than, say, walking? Let's put this to the extreme, why do we feel discomfort or excited, when jumping out of an aircraft? After all, the inputs match up: we have vision, we feel the acceleration (or actually, the lack thereof when jumping from the plane), we feel the air drag and we hear it too. Last but not least, we anticipated it (it was your own choice, nobody pushed you). The answer is simply that we are not used to it. Other than a counter-intuitive mental state for the case of jumping (our instinct didn't quite teach us that it would be a good idea to jump several kilometers down), our bodies are physically adapted to the presence of a ca. 10m/s^2 downward acceleration: our gravity. You can thank evolution for this. For the same reason, we have no problem being around in a pressurized environment of ca. 100MPa: our atmospheric pressure (rather the contrary, you'll not feel at comfort at all if you take that pressure away...). It exerts a force similar to the Earthly weight of ca. 10 Renault Twingos per square meter of surface, but our body does too in the opposite direction, and our touch sensors are trained to ignore their mutual presence. Same for acceleration: not only does the orientation change, but also the amplitude (0 for a free fall and >1 when accelerating a car). Worse than the difference, is the change: the vestibular organ is not particularly keen on change of acceleration, known as jerk.
For your interest: 
That said, whereas the human body has gone through a long optimization process as caused by evolution, it is still very well capable of adapting over a much shorter time interval than that of a life time, within a certain window of capabilities. Astronauts are capable of adapting to a 0G environment after a while (although it is not healthy for them), and you might be familiar with the human's capability to develop sea legs when sailing on a boat for long enough, etc.
* I cannot be held responsible for any effect this experiment might have.
A: I’m not 100% sure about this, but I think that in your example with the car you actually feel not the acceleration or deceleration as such, but the change of the acceleration or deceleration rate. If you would drive with a constant acceleration on a straight line for a while, you would adjust to it and feel it in exactly the same way as you feel the gravity.
A: You can not feel gravity as such because, as you said. it acts on every atom in your body. What you feel is the stress set up in your body when an external contact force (effectively the electrostatic force) is applied either to accelerate you or counteract the gravitational force.
A: You do feel gravity the same way as acceleration. That was Einstein‘s Point with the elevator thought experiment. The reason you cannot tell the difference between being in a rocket or a closed room on earth it’s because they feel the same.
A: I can think of four reasons why a car's acceleration feels different:

*

*Its changing. You're used to gravity and so it feels "normal" and you don't notice it. A car's acceleration changes as it goes into and out of curves, and so it's a constantly new stimulus, so you notice it more.


*It's directed in a non-downward direction. You're used to gravity acting downwards, but you're not used to a sideways force, so it stands out to you.


*The cues from your other senses make it more noticeable. Gravity isn't tied to motion, but a car's acceleration is. When you both feel acceleration and see yourself moving, it registered more strongly.


*The total acceleration is higher. You're used to a certain amount of "weight", and your effective weight is higher when you're experiencing both gravity and a car's acceleration,  and you experience the additional acceleration as being notable.
A: gravity and acceleration feel the same to humans.
so you do in fact feel it.
