Isn't according to GR, a frame attached to earth a noninertial frame? In this note of Leonard Susskind's GR course, he considered two different frames $z$ and $z'$. The $z$-frame is attached to the earth and the $z'$-frames moves up with an acceleration $g$. See equations $(1)$ and $(7)$.
In general relativity, we learn that a coordinate system attached to the earth is like an accelerating frame (ala principle of equivalence). But in this derivation, Susskind assumed the coordinate system $z$ is inertial and uses Newton's law for this frame. This is not uncommon. In Newtonian mechanics, we frequently take the observer fixed w.r.t earth to be inertial.
But is it not a contradiction? He is declaring a frame to be inertial when GR says that the frame attached to earth cannot be inertial. Please forgive my conceptual understanding. I am mostly using online materials to learn GR conceptually.
 A: You are right; if you stand on earth, you are not an inertial frame.
However, I think he is just trying to make a point about the relative acceleration of the elevator w.r.t the person standing on earth. If the relative acceleration is $g$ then the force experienced inside the elevator looks as if one would be in a gravitational field with gravitational constant $g$, i.e. acceleration and gravity are equivalent.
You might just assume that in his picture earth has no gravity, or you could add earth's gravitational force to his equations and the consequences would be the same (namely that the term generated by accelerating the elevator looks like an additional gravitational force).
A: In theory, a frame is either inertial, or it isn't.  In practice there are varying degrees of inertial-ness.  If the sources of error caused by omitting the fact that a frame is not inertial are small enough compared to other effects, we tend to handwave it away.
We quickly recognize that the frame of an observe on a merry go round is not inertial.  Its easy to observe this, because the centripetal and coreolis effects are non-trivial.
For most purposes, we consider the frame of someone standing on the ground to be inertial.  When aircraft simulators model the equations of motion, the North East Down coordinate frame is often treated as "inertial."  The non inertial terms, such as the Coreolis effect are small compared to the effects of winds, and you typically have a guidance system correcting for errors, so you get away with ignoring them.
For satellites, these effects are no longer ignorable.  Indeed, it is these effects which keep them in orbit!  If you are working with satellites, you often use an ECI frame (Earth Centered Inertial) like J2000 which doesn't spin with the earth.  ECI is close enough to an inertial frame that we even find the word "inertial" in its name!
Of course, it isn't inertial.  We're orbiting the sun, and that rotational motion creates non-inertial effects.  They are tiny compared to other gravitational abnormalities in the satellite scheme of things, but if we start talking about traveling to Mars, we need to account for our rotation about the sun most assuredly!  So there's a barycenter-centric coordinate system which is inertial, surrounding the center of mass of the solar system.
Okay, I lied.  It's not inertial.  We are rotating about our galactic center at about $7.96425522\cdot10^{-16} \frac{\text{rad}}{\text s}$, meaning we complete a revolution about once every 250 million years.  This is mind-numbingly close to an inertial frame, but clearly it isn't perfect.  We have to consider that rotational effect.
And at some point it should be clear that it would be egocentric to assume our galaxy isn't moving in some non-inertial way.  Those calculations can get quite tricky, as now we're on a scale where the inflation of space starts to play a significant role.
So, in summary, we call a frame "inertial" in a practical setting when its non-inertialness is so minor that we can ignore it for the discussion at hand.  This is certainly what Leonard Susskind did.
But, if you want a formal argument upon which you can rest, he happens to use the "Z" component in his vertical scenario, and specifies the he's only thinking about one dimension.  In ECEF (Earth Centered, Earth Fixed), and most ECI frames, the z axis is aligned with rotational axis of the Earth.  So he picked the one named axis where these frames are as inertial as they come!
A: The situation in GR is very simple: an inertial frame is a free floating and non rotating frame . Free floating or free falling  means that you do not have rockets or any kind of support attached to your lab,  and non rotating means that it is not rotating with respect to gyroscopes attached to the frame. This is the operative definition.  The formal one is this: In inertial frames the law of inertia holds: objects not subjected to forces move in uniform motion. In newtonian mechanics the definition is the same : inertial frames are those where the law of inertia holds.  What is then the difference?  It rests on the classification of gravity as a force. In GR it is not a force.  So in GR an Earth lab is non inertial because apples do not float, they fall, even if there is no obvious "force" acting on then (remember gravity is not a force) ,  while in newtonian mechanics it is inertial because apples fall due to the "obvious gravity force",otherwise they would float. In practice in GR all the inertial frames are small in terms of space and time. That is why they are called local inertial frames LIF. But you will see this later.
