As Pink notes below, it is possible to do a home artificial
insemination - and is probably more common than one might think. I've
found a few pages for you on the 'net that have instructions. The
turkey baster method is the most (in)famous - but there are several
others as well. Remember that basically the point is to get the sperm
into the uterus/fallopian tubes, and that the little guys help out by
swimming with any current they're given, so mostly what you have to do
is get them going the right way. Not that you're guaranteed success!
FertilityPlus: At Home Insemination
Instructions for three different methods; "turkey baster" (they
recommend a needle-less syringe), diaphragm, and diaphragm with tube
From a local TV station news report!
Malpani Infertility Clinic: Self-Insemination (DIY Insemination)
This goes into details of both the syringe and cup methods - and you
can purchase supplies if needed.
I found these three websites by doing a search on Google for the
keywords insemination and baster (yes, I'm a bit embarassed, but
sometimes the simplest searches are the best!). This search found me
over 600 hits, most of which are anecdotes about getting pregnant. I
wish you luck, and welcome requests for clarification.