An object which has been made at home can be called homemade. While this term is often used for cooking, it is used for many other things.
If someone were to ask you where you got that "thing", you can tell them they can make it themselves at home; it's homemade.
A synonym of homemade is homebuilt (alternatively home-built or home built). There are many references to things that can be constructed at home including homebuilt aircraft, cars, guitars, and homebuilt machines in general.
Another term that is not as widely used is homebrew. The term originates in brewing beer or spirits at home, but has been adopted for use in other fields, including electronics projects (here and here), computers, car projects, and a homebrew nuclear reactor.
DIY or do-it-yourself has become a very common term in the U.S., contrary to some comments here. We have the DIY Network, part of Home and Garden TV, which has become an institution among DIYers, handymen (and women), homemakers, and homebrewers.
To make something from readily available materials is to improvise. Improvising doesn't have to be done at home. It can be done anywhere. Here is a YouTube video example of improvising.