What is the weight of earth?

As odd as it sounds, the weight of earth is exactly zero, because the earth is in
orbit around the sun, and as such, the earth is free falling in space around the
sun.¹ Any object in free fall in space, including an object in orbit, is weightless.
That is why astronauts are weightless when in orbit around the earth.

Weight is a characteristic of an object as it relates to the gravitational field it is
resting in. You would have to take the earth to a much more massive world, like
Jupiter, and ignoring the difficulties caused by the gaseous make up of the
planet, put the earth on a rather large scale to see what the earth weighs
there.

The mass of the earth is another matter. The mass of the earth is 5.9736×10²⁴ kg,
or about 5,973,600,000,000,000,000,000,000 Kg.

_{1 An orbit is a special case of a free fall condition. As the orbiting object falls downwards, it
also travels transversely (sideways) at such a rate that its falling trajectory projects a curve
that always remains the same distance from the planet's surface.}

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Urgent bulletin/update:

After reading the above discussion, I went to my laboratory and performed an
experiment. I had heard somewhere that gravitational forces always occur in
pairs, between the centers of two masses, that the two forces are equal, and
that the force between me and the earth is what I call my "weight". If this is
generally correct, then the weight of any object depends on the other object
to which it is gravitationally attracted at the moment, and if that's true, then
I can weigh the earth on me.
In my laboratory, I placed a tiny mirror on the floor. I then took a bathroom scale
out of a cabinet, inverted it, and placed it top-down on the floor, with its digital
display visible in the tiny mirror. I then alighted upon the scale, placing my full
weight upon the surface that is normally the bottom of the scale ... the surface
with the label, the rubber feet, and the battery door on it. In this way, I was
able to weigh the earth in my gravitational field, and ... just as Sir Isaac might
have predicted ... it was precisely equal to my weight when measured in the
earth's gravitational field.
It would seem that in order for any contributor to accurately quote the earth's
weight, the question must specify the other object to which the earth is being
gravitationally attracted, and must also specify the other object's mass, and the
distance between the centers of mass of the earth and the other object. As any
of these details changes, so too does the earth's 'weight' change !