That's related more to induced current effects in the shielding material, and is more of a concern at higher freqs. For low freq and DC circuits we worry about lower freq mag field problems, say at 100kHz and under...although noise injection can come from anywhere.
You use steel and / or MuMetal or whatever material you've got that has high permeability -without saturation- to protect your circuit from external mag fields. In other words you need to steer the magnetic field -around- your circuit, so you give it a very easy path to follow in its own contained path around your device. If the material is too thin it is likely to saturate if the mag field is strong enough, and then it's not doing much. Once that shield material saturates, it's not steering the mag field anywhere in particular.
Depending on the local field strength and frequency you use the combination of shielding materials that works the best for best profit and effect - For instance you might find that a small steel box (a few dollars at the hardware store for a junction box for instance is a simple test, much heavier gauge than a candy tin) might give you enough noise attenuation for your lab situation. Or a double shield box design. Or you might need an annealed deep drawn MuMetal can that will shield your circuit all the way down to DC. Copper and Aluminum can shield at higher freqs. and more for RF issues. Sometimes a simple steel box with an overwrap of MuMetal foil works well enough at low freqs. If you don't kink the foil in a really sharp bend that MuMetal will still work surprisingly well without any heat treatment at all - and is miles better than nothing.
It just depends on what you're shielding against, for your situation and budget. You also look at protecting and shielding all leads (Chokes - ferrites usually) and equipment around the sensitive circuit - mag shielding is only part of it - you have to look at the noise delivered into your system from mains power and noisy high impedance grounds, noisy equipment in the lab etc.
The main thing is to keep your current loops with as small an area as practical, and try to minimize mag field interference as much as possible that way. Twisted short & shielded connection leads, for instance. Normally we'd never be using longer banana-plug and clip-lead style connections at low ppm for serious measures, but we see that all the time with the informal hobbyist setups - and those standard, separate red/black test lead assemblies are usually much too long and wind up acting as a pretty good loop antenna (transmit and receive) more than anything.
Do whatever works best for your situation, budget and measuring uncertainty goal.
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