I’ve always removed the hard drives when I disposed of a computer.
You have a town? 
Our county (mostly agricultural) accepts electronics either at the recycling centers where we bring our cans, bottles and paper, or at their periodic hazardous waste collections. You should be able to find out about it on the web site for your county.
When we get rid of computers, we pull the hard drives first. I suppose we’ll eventually bury them all somewhere on the property to bemuse and intrigue future generations.
Put it on eBay, for parts.
I put a screw through my last hard drive.
Even my little county has electronics recycling, but in years past it’s been only at the spring cleanup. Now we have a small recycle center that will take them year round, assuming you can find someone to pay the fee to.
Harddrives are 6" x 4" x about 1" They have gotten thinner over the years.
The cables can be pretty stuck on old drives, so just cut the wires off.
Then open the drive itself with a small screwdriver and rip out the guts with a claw hammer, breaking the disks / platters as you do. You can throw the platter pieces in the trash.
I’d bet that your county has some program for old electronics as the EPA fines can be pretty steep. Leave the PC next to your garbage. If they don’t take it, call the collection Co and ask what to do.