Speed of light = 3 x 10^8 m s¯¹ = frequency x wavelength. So if the frequency = 10^7 Hz, the wavelength = 30 m. One-tenth of that is 3 m. If the diagonal is 3 m, the sides of the square are 3/√2 ≈ 2 m.

That is where the numbers come from, the 'why' is more interesting. To block a transverse  EM wave we need to present it with a short circuit plane - we need to connect together regions of space that would otherwise support a voltage gradient. This could be a continuous sheet of metal of conductor, but it could be reduced to just joining a few nodes to connecting regions in space that would be at different voltages if the wave was present, and if we do this will prevent the field building.
This use of mesh instead of sheets is like approximating something curved by a series of flat pieces. How badly connected those flat pieces can be before the 'short circuit' is not good enough to suppress wave propagation  can be seen as an example of sampling theorem. The tenth of a free space wavelength is a bit of a rough rule of thumb, it could be 1/8 or 1/20, and the degree of attenuation will change - getting better with more metal and less holes, but much by the time you have an aperture of half a wave, you can propagate a full strength wave through it - Nyquist.

Or if you prefer, you can sample a sine wave at 10 dots per cycle it looks like a wave if you sample it at only 2 dots per cycle it looks like DC.

Mike

A nanosecond is about 1ft in free space.

Thank you very much Chris and Mike. So, two things; 

1. How do you predict the attacking frequency?

2. If the concept is enhanced by reducing the mesh size (I think that is what Mike suggested), could the rebar of a poured concrete slab be used or is ferromagnetic material out?

1. How do you predict the attacking frequency?

Know your enemies, try and capture a sample of their equipment. Or place a guess that it will be short wave or higher, as to radiate lower freqs / longer wavelengths needs massive antennas - either deliberate ones, or accidental ones. Many buildings and bits of kit have wires metres long that act as accidental antennas in the VHF, some buildings have wires tens of metres long that act as accidental short wave antennas, and very few have  wires hundreds of metres in extent that would radiate well on medium wave.

EMC standards reflect this fact, (which relates to the fact the humans are meter scale animals and therefore usually build on a scale that is fractions of a metre to tens of metres,) and require radiated tests from 30MHz upwards (10m waves and shorter) , but assume that anything slower (longer waves) is likely to be conducted up a wire and amenable to filters.

The saving grace at the other end is that higher frequencies are harder to generate by accident, so are less likely to be flying about, and generally accidental radiators  roll off above a GHz leaving you with intentional radiators like only comms systems and RADAR kit to think about  (maybe the odd leaky microwave oven on 2.4GHz) and normally you know where they are.

2. If the concept is enhanced by reducing the mesh size (I think that is what Mike suggested), could the rebar of a poured concrete slab be used or is ferromagnetic material out?

And yes rebar weld mesh of half inch steels on 6 inch centres is very good as a screen for freqs up to about 100MHz, so long as you can weld onto the edges and or bend it about to make the box continuous.
Chicken wire is good up to UHF (say about 500MHz),again the problems are the joints.~ finer meshes are sometimes used as a material for the reflector dishes for field deployed parabolic dishes - as well as being easier to carry to site they are better in the wind as well.

The facility we have at work for our high energy pulse power tests has copper mesh observation windows of 1.5mm weave pitch, that do start to  leak a bit by you reach 10GHz (3cm) but if you are silly enough to get in and shut the door, no mobile phone or walkie talkie signals will get in or out. (GSM handset signals ~ F= 900MHz wavelength about 1ft some of the newer 3/4gStuff goes  up to 2.1 GHz waves of 15cm or so) 
If you ever have to design something like a nuclear bunker you end up with mesh in the air ducts and that sort of thing, and every where else fully welded steel floor, wall and ceiling panels, and spring seals on the doors.
If it really needs the ultimate and you need a decent flow of air or water so a solid panel with some pinholes will  not do, then honeycomb  welded panel, is the way forwards and electrically looks like an array of tubes, any one of which is too small to permit significant radio wave propagation. In this case they start to leak when the radio half wavelength approaches or is less than the guide largest interior dimension, and they begin to behave like waveguides

M

I have seen a room being prepared for the installation of an MRI scanner. It was lined with copper, so it was like standing in a still. So would that have been to keep stray magnetism in, or interference out?

when you lie in an MRI scanner then you are 'illuminated' with a very strong static magnetic field - so nearby steel objects are banned - indeed folk have been badly hurt by flying oxygen bottles and so on  in the early days when folk have not been thinking - now they only allow non-magnetic bottles and things in. But once the atomic nuclii are all facing along the direction of that field, we 'twang' them with about a kW or so of RF, and listen for the various frequencles that they then spin at, with radio receiver to detect weak signals that are pretty close to the theoretical limit of what can just about be received. So the screening keeps both the RF excitation ("pump frequency") in the box, so it does not jam other electronics in the building, and at the same time also protects the detector/ receiver from what is outside, and if it was not for the big magnet steel would be OK, but it isn't making things like transformers in the power supplies very tricky.

The fine structure in the received spectrum allows the concentration of various sorts of atoms, and what they are bonded to to be deduced (as they are detuned by the magnetic fields of the near neighbour atoms.)

Having designed some electronics for just such a situation, I will say that it is very challenging, as all sorts of common parts, not just iron or ferrite cored inductors but also a lot of 'innocent' looking parts like connectors fixings, mechanical parts. do actually have to be specially made in a non-magnetic variant (no nickel, no iron, no cobalt etc).

The kW of RF is why you need to be cooled while you are in the thing as inside the magnet tube is the big antenna array that is both illuminating you and listening for the nuclear magnetic resonances from the atoms inside you - it is really clever.;-) The radio frequencies, the local magnet strength, and how the antennas are used, can be scanned, swept  and configured to allow regions of interest in the body to be selected with remarkable precision.

This scanner explanation is a bit of a simplification, so apologies to the many folks whose life's works are in scanner design, but the bit about the screening, what it is for and why it is not steel,  is precisely accurate.

Mike.