The biggest problem with giant anything, as everyone else has mentioned, is pressure. You can't scale up a human, for instance, because human bones couldn't support the force in play. That means the mech has to be built with strong materials, and the mech has to be lighter than a scaled creature.
Bones and armor
Titanium is a good choice, though it could get expensive for giant mechs; granted, if you're building an enormous machine, cost probably isn't a huge factor. Titanium has a tensile strength of about 434 MPa (63,000 psi) - about the same as steel, but for much lower weight. Titanium alloys have a much higher tensile strength, over 1400 MPa (200000 psi). Regardless, it would make a good armor as well as good support. As a bonus, titanium is very corrosion-resistant, which means you won't have to worry about your robot rusting.
Carbon nanotubes are also a good bet. We don't have the technology to make anything large-scale with carbon nanotubes, but we'll get there eventually. There are many types of carbon nanotubes, depending on how the atoms are arranged; one had the tensile strength tested at 63 GPa (9,100,000 psi). Other varieties are theorized to have a tensile strength of over 160 GPa (23,206,000 psi)! However, as strong as carbon nanotubes can be, they are strongest when pulled, not pushed, and are weakest against forces from the side. They would make great "bones", but probably not outer armor.
Graphene is super strong - a tensile strength of 130 GPa (18,855,000 psi) - but is very brittle. It would make an excellent bone.
Oddly enough, bamboo may actually work quite well. It has a tensile strength of 350-500 MPa (50,760-72,520 psi), and is quite light, with a lower density than even Graphene. Bamboo is also very easy to come by; it grows quickly in natural environments, with no need for mining or creating in a lab.
Other materials may work as well, to varying degrees; I'm sure the future holds much stronger materials, too.
Why not lighter?
The reason titanium is a better choice than steel is weight; but what if we make the mech even lighter? A giant robot that is essentially a thin shell may not stand up well in hand-to-hand combat, but it would satisfy the requirements!
A mech built using thin plastic or aluminium tubing and steel wire as bones and muscles could but placed inside a canvas balloon and inflated; the canvas would be nearly immune to 'crushing' attacks, and the bones wouldn't have to support much weight. If the mech were filled with a mixture of air and a lighter-than-air gas such as helium, it would be able to support itself with very little effort. Because of its low weight, the internal structure would be very thin; a giant mech made of inflated canvas could be built today for less than the price of a used mid-sized car.
Put it together
I suggest the best of both worlds: an air-filled canvas robot, plated with titanium, and with carbon nanotube bones. The internal structure would be filled with a lighter-than-air gas, resulting in a machine that towers above the earth, but can step on a house without crushing it. If it wants to.
As an added bonus, there is a lot of extra room in the mech that can be used to store weapons or snack food, and the bouncy exterior would protect any occupants.