I was able to swing $230 total for two x230s with the base i5 and ssds last fall/winter (actually three, but the third doesn't really count (arrived damaged, seller initiated a refund then ghosted, got a full refund from ebay)). If you're really lucky, you can find a computraced one for super cheap ($85 in my case) and then coreboot it.

That said, x220s and x230s seem to have gotten more expensive since then, I haven't seen any similarly good deals lately.

Check the buyer’s guides on r/thinkpad. Typically you’re looking at T or X series, with T giving the most flexibility. T410 to T430 can be had for less than $200 and should be fine for server use. The numbering scheme is Txy0, with x the screen size (4 for 14), b the Core i generation (0 is core 2 duo).

As someone who's not a huge fan of used laptops as servers (they've already seen hard use and are not really meant for 24x7 operation either), consider the Odroid H2+ instead: https://www.hardkernel.com/shop/odroid-h2plus/ Tiny, x86-based, and only $119 plus RAM and storage.

Yeah I’m in the same boat right now. I want an always on machine to connect a disk to that doesn’t draw too much power. I looked at NUCs but they seem far too expensive for my usecase.

I use a powered USB hub and it got rid of my flakiness problems.

I went for a ThinkPad 13. Add on the dongle for an RJ45 port (about GBP10) and you're sorted!