I'm wondering if there is any way to create adenine, thymine, cytosine, or guanine from chemicals available to anyone, using basic lab equipment.
Based on jonsca's comment, extract DNA from something. The protocol will involve detergent and isopropol alcohol. The details are available in several places and is fairly simple, but you won't get much DNA per onion, so be ready to prepare a lot of them. To do the degradation, there are several things you could try. If your detergent contains nucleases ( many contain enzymes but do a poor job of labeling which enzymes ) you could mix that in and react it at 37C for a while. Can also try high pH solutions, if you happen to have NaOH around. If you do the DNA extraction differently, you can get the RNA instead of DNA. RNA is very unstable at high pH, and RNAse enzymes are EVERYWHERE. So the RNA should degrade into nucleotides pretty readily if you just use sloppy lab technique. When I extract RNA in the lab, I use phenol/chloroform to pull the proteins and genomic DNA away from the RNA, which stays in the aqueous layer. You probably don't have access to those at home, but cooking oils may be hydrophobic enough to pull it off. Also, it helps to have a centrifuge.
Don't ask me how to purify the nucleotides from your solution and ask me even less about how to verify what you have at home. Verification would require something like a UV-Vis or HPLC or NMR. Good luck getting those at home.