Fanless Mini PC HTPC Intel Pentium J2850 Quad Core 2.41GHz Micro PC Barebone - ebay $150 USD w/$50 s&h
My comment: It's a Pentium, quad-core, 2.41GHz.
One (empty) ram slot (probably doubles the total cost including a separately purchased 4GB ram stick).
Spec sheet beneath picture gives more details:
-supports Win 7 & 8, up to 256GB SSD HD, & etc.
I wonder if this means it will not accept my external 200GB EIDE HD, or 500GB & 750GB SATA HDs?
edit: 'Compatible with Win 7 & 8' implies 'incompatible w/Win 10 & etc'.
Even though it is a "small" PC, put in as much RAM as you can afford (8GByte or 16GByte) as that will extend its useful product life significantly. The problem is that your EIDE hard disk cannot be connected to this main board at all. Actually, EIDE isn't supported on any new mother boards for several years now.
Your SATA hard disks could be connected and would work if these didn't have too much storage capacity for the onboard SATA controller. Also, it is highly likely your SATA hard disks are 3,5" models. This model requires a lot more power to operate than the 2.5" model. In other words, your hard disks are too big in size (both physically and in capacity) and too power consuming. The power supply that comes with this device isn't able to cope with those power requirements.
Low power devices always sacrifice (a lot of) capability to get those low power numbers.
You should buy the biggest SSD hard disk this board supports, again to extend the product's life.
If you do all of this, you will be very pleasantly surprised about this device for a long time to come. However, it won't be cheap to buy. It will be quite cheap to run.
Or buy a cheap PC case with power supply, transfer your old main board into this case, including your EIDE hard disk. Sell it / donate it / use it as a NAS server (you should get decent power supply if you do this).
Add new main board with new processor and DDR3/DDR4 RAM to your old PC case. A decent Asus main board with decent Intel i5 processor and 8GByte of DDR3 RAM will set you back around 300 USD here in Paraguay. And in the U.S. hardware is cheaper anyway. You can still use your SATA hard disks on this new main board, the onboard SATA controller can slow down to SATA 1 speeds if your SATA hard disks are SATA 1 models (which is very likely in an old PC as yours is).
If you can do the labor and hunting for deals on computer parts yourself, building is usually the cheapest option to buy, but not necessarily cheapest to run. And if not, you'll pay for the labor someone else did in getting parts and assembling them for you.
While I do understand why someone buys an off-the-shelf PC for home use, you won't see me do that. I (still) like to build my own machines from scratch. Mainly because I don't want to be stuck with a sub-par PC that someone else deemed good enough. And that is even more true for laptops, if you talk about sub-par computing (but that is a rant for another day).