Do-able, but you are probably going to be going with cheaper (low quality) tools. A better quality midi lathe will run you $800 alone. That said, you can go with something cheaper like the Harbor Freight 10x18 which is $260, but it has some limitations (only 5 speeds and belt driven, although I think belt drive is better when drilling on the lathe). The jet 1015 has 6 speeds (including 500 rpm which is better for drilling compared to 750 in my opinion) and is probably a little better quality, but it runs closer to $500.
At a minimum, you'll need a dead center, live center, digital calipers, lathe tools (carbide or HSS, a single budget carbide tool to start with is probably cheapest), method for sharpening HSS tools (stone is fine for skew and scrapers, but grinder is more appropriate for gouges), scroll chuck and jacobs chuck (if drilling on lathe), pin jaws or pen jaws for scroll chuck. My guess is that this list is about $350 depending on what you go with, but there is a lot of variability. You could blow $350 on a single Starrett digital calipers or get 'good enough' with the $15 ones. The scroll chuck with jaws is probably one of the bigger tooling expenses, I like my Nova G3, but you are looking at probably $220 or so for the chuck, adapter, and jaws.
Then there are the "consumables": glue, sandpaper, kits and blanks, drill bits, bushings/mandrel (unless you TBC), finishes. This might be $50 or $100 just to get set up for your first pen.
So, in summary, the cheapest I think you could do at this time if buying new is: $260 for lathe, $350 for tooling, $75 for consumables. So you might be able to sneak in under the $800 mark, but you are going to sacrifice at least some on quality of tools and 'ease of use' or functionality.