I want to use a Raspberry Pi, along with an external USB hard drive, as a miniature SSH server, but I don't have a monitor or TV to use with it. It isn't worth it to buy a monitor/TV for the set up, since once I have the SSH set up I won't need any output (I could do the work remotely) but I don't know of any other way to see the output.
You don't need the system running to configure sshd, and I think the stock raspbian (and probably other distros) will start the server by default. They certainly will have it pre-installed. So you can, for certain, set this up without the pi, meaning you don't have to worry about the monitor.
Mount the SD card on your computer and configure /etc/ssh/sshd_config to allow root login, add your public key to $HOME, etc. That's the easy part, presuming you know how to configure sshd.
Next, check that sshd is actually set to start in /etc/rc2.d (runlevel 2 seems to be the default on raspbian, although it seems like 3 to me, but those directories are more or less the same). If you see
@S03ssh or something in there, you are good to go. If not, you can just add such a symlink (the target is
/etc/init.d/ssh), which will work although you may want to do a more proper debian services thing once the system is running.
ln -s ../init.d/ssh S03ssh
Put the SD card and an ethernet cable from your LAN into the pi, then plug in the pi power. Wait a minute or two and ssh in. If you need wifi, you'll have to do more configuration on the card first, so hopefully you have a cable.
If you don't get some part of this, ask specifically and I can explain in more detail.
The short answer is no, at least not in the way you're thinking of.
It's true that one could use an HDMI capture card (Amazon has several) to capture the Pi's output and display it on the computer, but this route is likely more expensive than just buying a cheap monitor or using a monitor that you already have.
If the problem is that the monitors you have don't have HDMI, then you could use an HDMI-to-DVI cable (like this one) to hook up the Pi.