There are several such 0G operations; the "commercial" ones (such as the US 727 operator) are separate from the better-known ones operated as part of the various space programmes. There's a little known one under the latter category using an A300 in France, part of the European Space Agency.

In aircraft terms it's all straightforward stuff; the aircraft is not unduly stressed at all and well within it's normal structural operating regime. Such operations do not do many flight hours in a year, which is why they tend to use older, cheap to acquire aircraft like the 727 where fuel consumption is not an issue. If I recall correctly they need some minor mods to things like fuel/oil pumps to keep them running under sustained 0G conditions.

The 727 operator is I believe the same one that offers them to skydiving clubs for the chance of jumping out of a jet the same way that D B Cooper did it, down the rear ventral stairs that a 727 has.