A wire feed welder would be too complicated to make at home, but, over the years people have published articles on how you can build your own stick arc welder. A very popular Instructable
tells how to make a 120 volt AC arc welder from two microwave transformers. See also my previous Instructable for links
to three sets of plans for building an electric arc welder. Here
are some instructions on how to use an automobile's alternator to weld for emergency repairs. Another link there
has an additional scheme for using your vehicle's alternator for welding. And, if you scroll down, that link also includes some brief instructions for linking multiple car batteries together to weld. This 1980 article from Mother Earth News
tells how to make a very portable DC arc welder from an alternator, a deep cycle battery, resistors, and an old lawnmower
. This unit produces about 50 amps, and will be limited in what it can weld. There was a time when I was so eager to build one of these, but I never did. Here is a video
of a more refined welder built on a lawnmower frame. Despite the 50 amp. output of the Mother Earth News
welder, a welder similar to the one in the last link handled 3/32 inch rod fairly well.
A used welder can be a good buy. Watch estate sales and other auctions, as well as Craigslist for your area. I bought my 230 volt Miller Thunderbolt stick welder on eBay. The photo was muddy-looking and no one else bid on it. The very good auction price became a quite reasonable price after shipping costs were added. It has been a great welder. I bought my Hobart flux core wire feed welder as a factory refurbished unit. No one has much good to say about cheap import welders. Owners are usually soon disappointed.
(The photo is from Google Images.)