I have some old micro SD cards that I'd like to use if possible. How much space does Raspbian need and how much room should I leave over? (What is the minimum size card I can use?)
For jessie (full) I'd use a minimum 8GB SD card, that should leave 2-3 GB spare.
For jessie lite I'd use a minimum 4GB SD card, that should leave a GB spare.
While what @joan replied suffices what you asked, you may still consider other things like what other software you wish to install. If you do some RnD, you might need more space for some data if you are not using some USB drives.
So, just be considerate about these.
Absolute Minimum card size is 1GB for a base Raspbian system with no GUI.
If you're using a USB Hard drive or USB Stick as the root file system, you can get away with an even smaller one.
I installed Jessie Lite (2017-03-02-raspbian-jessie-lite.zip) on a Pi 3 with a 2 gigabyte micro SD card. Here's what df showed me:
Filesystem Size Used Avail Use% Mounted on /dev/root 1.8G 846M 807M 52% / devtmpfs 459M 0 459M 0% /dev tmpfs 463M 0 463M 0% /dev/shm tmpfs 463M 6.2M 457M 2% /run tmpfs 5.0M 4.0K 5.0M 1% /run/lock tmpfs 463M 0 463M 0% /sys/fs/cgroup /dev/mmcblk0p1 63M 21M 42M 33% /boot
So it looks like 2G is about the lowest you can go with a stock Raspbian Jessie Lite image.