You can always leave to other country, or minimize your contributions with creative accounting.
Also very common is to build your own house by yourself DIY style. It is significantly cheaper because you do not have to pay taxes from work you do for yourself.
The United States taxes its citizens' income worldwide, at least above a certain income threshold, and from my understanding requires yearly filing regardless. IMO it's unlikely that simply moving to a different country would allow you to dodge a federal mandatory pension (provided it happened in the US).
Social security is the United States' federal mandatory pension, and it does not generally apply to foreign income (unless in a country that also has a mandatory pension with which the US has a tax treaty).
Oddly enough, you can opt out of the Social Security - some employees of government agencies don't have to contribute and some religions are also exempt.
Personally, I feel that anybody should be able to permanently opt-out if desired.
Social security has an awful lot in common with a federal mandatory pension. If anything, it would probably be easier to opt out of a pension system (where pension implies that the individual benefit is more fully funded by the individual payments).
People that can show they are making payments into a similar program elsewhere often don't have to make social security payments.
Also very common is to build your own house by yourself DIY style. It is significantly cheaper because you do not have to pay taxes from work you do for yourself.