We can do some simple math here to see what our constraints really look like.
Let's assume the meteor is mostly made of iron, because I heard that's a common composition in a museum somewhere. Assuming spherical shape, it would be around 3.3 * 10^16 kg. The kinetic energy of the meteor would therefore be 5.3 * 10^24 Joules. We'll assume this is the Kinetic energy at contact with zero gravitational potential energy left. Out of curiosity though, since the escape velocity of Earth is 11.2 km/s, 11.2^2 = 125 and 18^2 = 324, and so gravitational potential energy accounts for at maximum 38% of the total meteor energy.
Now the one fundamental thing you have to keep in mind here is this: If you really really really want to stop the meteor instead of deflect it, then that is the raw amount of energy you have to deal with. You can dissipate it, you can reflect it, you can eat it (destruction), but no matter what, that is your the raw quantity of your challenge. This is a lot of energy. It is equivalent to 25 million Tsar Bombas. How do you absorb the energy of 25 million Tsar Bombas with a net? <PERSON> even describes this amount of energy as 11 times the estimated energy of the Chicxulub meteor impact that destroyed the dinosaurs (granted, our calculations are very rudimentary, e.g. sphere of iron). How do we stop 11 Chicxulub asteroids with a net? This is the question you may as well be asking.
Thankfully though, you have a very accommodating budget. So let's think of a solution!
Again, to emphasize, we have to dissipate energy. When you catch a baseball in a glove, or hit a tennis ball with a racket, the energy is dissipated through sound waves, stress, strain, and friction heat of the catching device. In more extreme catching scenarios, the catching device is specifically designed to break in intentional ways, in order to dissipate large amounts of energy (like table saws with flesh sensors or crumple zones in vehicles). 921
Probably our best approach for a net is to make it out of something really strong, but not so we can hold the meteor in the sky, rather so the meteor has to spend a ton of energy when it breaks the net.
Here is how I envision it
You make a net out of tons of carbon nanotubes. You dissipate the energy by planning on the carbon nanotubes breaking. I'm no mechanical engineer, but I'll see what I can do with rough calculating. Apparently researchers have developed high-strength carbon nanotube films with up to 9.6 GPa of tensile strength before breaking. Say in the next 50 years we get that to 10 GPa (being conservative). It looks like carbon nanotubes usually survive about 15% to 20% stretching before they break. So if we have a net that is carbon nanotubes in one single direction, in a 10km by 10km square, then if force is applied down on the net evenly (such that tension is the main force at play, and we ignore shear forces), we can find out how much energy it costs to break the net at any given net thickness.
* Pressure when net breaks = 10 GigaNewtons / meter ^ 2
* Force when net breaks = net thickness * net width * pressure when net breaks
* Energy when net breaks = force when net breaks * distance over which the force is applied
So
* Energy when net breaks = (15% * 10km) * (10 km) * thickness * (10 GigaNewtons / meter^2), or in other words...
* Energy when net breaks / thickness = 1.5 * 10 ^ 17 Newtons
And so if we want to stop a 5.3*10^24 Joule projectile, we need a carbon nanotube net 10 Km long, 10 Km wide, and........
36.... MILLION meters thick!
See here
I'm actually impressed!! My first edit of this answer was so incredibly hilariously wrong! at 18 m/s we need 3.8 meters thick nanotube...... but at 18 km/s we need a 36 million meter thick net! Again, at 10 km wide and 10 km long.... 36 million meters is 5.6 times the radius of the Earth.
You can also fine tune it, if you look at the formulas. If you increase the stretch distance, you don't need as thick a material. But it will simply never compensate for that raw amount of energy. This meteor catching exercise is basically futile. but fun in any case!
The way the terms work out, you can squint your eyes and make a different net depending on how you interpret it. 921