If the painting is too large to scan at home, you can have it done at a printing shop -- I think many of the smaller ones have been bought up by Fedex and Kinkos, but even they may still do it. They have the equipment to do it at very high resolution and also to make the print. You can find listings of print shops in your local area in the yellow pages and online and call around to see if they offer that service.
If you prefer, just have the painting scanned and get the image file, and there are many POD (print on demand) websites available. I have used
http://www.iprintfromhome.com/ (yes, that is really their name), with good results. They have a variety of different papers, and will send you a sample pack so you can see the different results you get on them. And if you're thinking of making these prints because you want to sell them, iprintfromhome will also ship your prints directly to another address (i.e. your customer) at no extra charge, though I have never used that service.
Another place to get prints made is Fine Art America (
http://fineartamerica.com/), though they are mainly meant for you uploading your image and other people buying it from them. However, you can also use it for buying the prints yourself. They do have a good image control system that prevents people from stealing your image from the website. A particular thing I like about that website is, if I want to make prints from something I have scanned myself, you can upload your print, and it will tell you the maximum size print you can make without loss of detail. (I make postcard size pictures, so obviously there is a limit to how much I can enlarge them and knowing that from a professional is important.)
Hope this helps!
